All practice questions

Question 2361Question →

In the standard $(x,y)$ coordinate plane, a square has two adjacent vertices at $A(1, 4)$ and $B(4, 8)$ . If the square lies entirely in the first quadrant, what is the $x$ -coordinate of the vertex diagonally opposite to $A$ ?

Show answer & explanation

Answer: 8

Answer

The

x

-coordinate of the vertex diagonally opposite to

A

is

8

.

The vector from

A(1,4)

to

B(4,8)

is

\vec{AB} = (3,4)

with length

5

. Since the adjacent sides of a square must be perpendicular and equal in length, the vector translation must be either

(4, -3)

or

(-4, 3)

. Applying the translation

(4, -3)

to

B

and

A

gives

C(8, 5)

and

D(5, 1)

, both of which lie strictly in the first quadrant. The diagonally opposite vertex to

A

is

C

, and its

x

-coordinate is

8

.

Step-by-Step Solution

1

Calculate the vector representing side

AB

and its length.

\vec{AB} = (3, 4)

, which has a length of

\sqrt{3^2 + 4^2} = 5

.

This determines the length and orientation of the square's base.

2

Determine the perpendicular vectors of equal length to find the directions of the adjacent sides.

The perpendicular vectors of length

5

are

(4, -3)

and

(-4, 3)

.

The adjacent sides of a square are perpendicular and equal in length to side

AB

.

3

Find the coordinates of the other two vertices for both possible positions of the square.

Position 1 vertices:

C = B + (4, -3) = (8, 5)

and

D = A + (4, -3) = (5, 1)

. Position 2 vertices:

C' = B + (-4, 3) = (0, 11)

and

D' = A + (-4, 3) = (-3, 7)

.

Adding the perpendicular vectors to the coordinates of the base vertices yields the remaining vertices.

4

Determine which square lies entirely in the first quadrant and identify the target coordinate.

The square

ABCD

with vertices

C(8, 5)

and

D(5, 1)

lies entirely in the first quadrant because all

x

- and

y

-coordinates are positive. The vertex diagonally opposite to

A

is

C(8, 5)

, which has an

x

-coordinate of

8

.

The alternative square

ABC'D'

has a vertex at

D'(-3, 7)

, which lies in the second quadrant.

Key Concept

Using vector methods to determine the coordinates of a geometric figure on the coordinate plane under specific quadrant constraints.

Estimated Time:2m 0s

Question 2362Question →

A commercial building has two water reservoirs. Reservoir $X$ contains $1,200$ gallons of water and is draining at a constant rate of $18$ gallons per minute. Reservoir $Y$ contains $360$ gallons of water and is being filled at a constant rate of $22$ gallons per minute. After how many minutes will both reservoirs contain the exact same amount of water?

Show answer & explanation

Answer: 21

Answer

The two reservoirs will contain the same amount of water after 21 minutes.

The correct answer is 21 minutes. By setting the expressions for the volume of both reservoirs equal (

1,200 - 18t = 360 + 22t

) and isolating the variable, we find

40t = 840

, which simplifies to

t = 21

.

Step-by-Step Solution

1

Translate the physical scenario for Reservoir

X

into an algebraic expression.

1,200 - 18t

Reservoir

X

starts with

1,200

gallons and loses

18

gallons per minute over

t

minutes.

2

Translate the physical scenario for Reservoir

Y

into an algebraic expression.

360 + 22t

Reservoir

Y

starts with

360

gallons and gains

22

gallons per minute over

t

minutes.

3

Set the two expressions equal to each other and solve for

t

.

1,200 - 18t = 360 + 22t \implies 840 = 40t \implies t = 21

Equating the two volume expressions allows us to find the time

t

at which the volumes are equal.

Key Concept

Translating and Solving Algebraic Word Problems

Alternative Method

Instead of solving algebraically, one can check the rates of change relative to each other. The distance between the initial volumes is

1,200 - 360 = 840

gallons. Since they are moving toward each other (one draining, one filling), their relative rate of convergence is

18 + 22 = 40

gallons per minute. Dividing the total volume difference by the rate of convergence gives

840 / 40 = 21

minutes.

Estimated Time:1m 30s

Question 2363Question →

A local coffee shop sells two types of coffee blends: House Blend and organic Reserve Blend. A bag of Reserve Blend costs $4.50$ dollars less than three times the cost of a bag of House Blend. A customer purchases $3$ bags of House Blend and $2$ bags of Reserve Blend for a total of $54.00$ dollars, excluding tax. What is the cost, in dollars, of a bag of Reserve Blend?

7.00

10.50

15.00

16.50

19.50

Show answer & explanation

Answer: 16.50

Answer

The cost of a bag of Reserve Blend is 16.50 dollars.

The cost of a bag of Reserve Blend is 16.50 dollars. Let

x

represent the cost of a bag of House Blend. A bag of Reserve Blend costs

3x - 4.50

. Since the total cost for

3

bags of House Blend and

2

bags of Reserve Blend is

54.00

, we write the linear equation

3x + 2(3x - 4.50) = 54

. Expanding the terms gives

3x + 6x - 9 = 54

, which simplifies to

9x - 9 = 54

. Adding

9

to both sides yields

9x = 63

. Dividing by

9

gives

x = 7.00

. Substituting

7.00

back into the Reserve Blend expression yields

3(7.00) - 4.50 = 16.50

dollars.

Step-by-Step Solution

1

Define variables for the costs of each blend.

Let

x

represent the cost of a bag of House Blend in dollars. The cost of a bag of Reserve Blend is represented by the expression

3x - 4.50

.

Establishing algebraic expressions for the unknowns based on the problem description allows us to set up a linear equation.

2

Set up the linear equation based on the total cost of the purchase.

The total cost of

3

bags of House Blend and

2

bags of Reserve Blend is

54.00

dollars:

3(x) + 2(3x - 4.50) = 54

.

The sum of the individual total costs of the two blends must equal the overall purchase total.

3

Distribute and combine like terms to simplify the equation.

3x + 6x - 9 = 54

simplifies to

9x - 9 = 54

.

Applying the distributive property removes the parentheses, allowing like terms to be combined.

4

Isolate the variable term by adding

9

to both sides.

9x = 63

.

To solve for

x

, we must first isolate the term containing the variable by performing the inverse operation.

5

Solve for

x

by dividing both sides by

9

.

x = 7.00

.

Dividing isolates

x

, giving the cost of a bag of House Blend.

6

Calculate the cost of a bag of Reserve Blend using the expression from Step 1.

3(7.00) - 4.50 = 21.00 - 4.50 = 16.50

dollars.

The question asks for the cost of a bag of Reserve Blend, not the House Blend, so we evaluate the expression

3x - 4.50

at

x = 7.00

.

Key Concept

Setting up and solving a single-variable linear equation to solve a real-world word problem with multiple unknown quantities.

Estimated Time:1m 30s

Question 2364Question →

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the point $(2, -3)$ and does not pass through the origin. The distance from the origin to the line's $x$ -intercept is twice the distance from the origin to the line's $y$ -intercept. Which of the following could be the slope of line $L$ ?

-2

-\frac{1}{2}

\frac{2}{3}

1

2

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

The slope of the line could be

-\frac{1}{2}

.

The correct answer is

-\frac{1}{2}

. By setting the

x

-intercept of the line to

(a, 0)

and the

y

-intercept to

(0, b)

, the distance condition gives

|a| = 2|b|

. Substituting the point

(2, -3)

into the intercept equation of a line,

\frac{x}{a} + \frac{y}{b} = 1

, yields

\frac{2}{a} - \frac{3}{b} = 1

. Solving the two cases (

a = 2b

and

a = -2b

) gives the intercept pairs

(-4, -2)

and

(8, -4)

. Calculating the slope

m = -\frac{b}{a}

for both pairs results in

-\frac{1}{2}

and

\frac{1}{2}

. Since

-\frac{1}{2}

is one of these possible values, it is the correct choice.

Step-by-Step Solution

1

Represent the line

L

using its intercepts. Let the

x

-intercept of

L

be

(a, 0)

and the

y

-intercept be

(0, b)

. Since the line does not pass through the origin, we have

a \neq 0

and

b \neq 0

.

The equation of the line can be written in intercept form as:

\frac{x}{a} + \frac{y}{b} = 1

Using intercept form allows us to directly relate the coordinates of the intercepts to the given point and the distance condition.

2

Translate the distance condition and the given point into equations. The distance from the origin to the

x

-intercept is

|a|

, and the distance to the

y

-intercept is

|b|

. The problem states that the distance to the

x

-intercept is twice the distance to the

y

-intercept, so

|a| = 2|b|

. Additionally, the line passes through

(2, -3)

, so we substitute

x = 2

and

y = -3

into the intercept equation.

|a| = 2|b|

and

\frac{2}{a} - \frac{3}{b} = 1

This sets up a system of equations to solve for the unknown intercepts

a

and

b

.

3

Solve the system of equations by analyzing the two cases for the absolute value:

a = 2b

and

a = -2b

.

Case 1: If

a = 2b

, then

\frac{2}{2b} - \frac{3}{b} = 1 \implies \frac{1}{b} - \frac{3}{b} = 1 \implies -\frac{2}{b} = 1 \implies b = -2

, which gives

a = -4

. Case 2: If

a = -2b

, then

\frac{2}{-2b} - \frac{3}{b} = 1 \implies -\frac{1}{b} - \frac{3}{b} = 1 \implies -\frac{4}{b} = 1 \implies b = -4

, which gives

a = 8

.

Resolving the absolute value yields the exact coordinates of the intercepts for both valid scenarios.

4

Calculate the slope

m

for both cases using the formula

m = -\frac{b}{a}

.

For Case 1,

m = -\frac{-2}{-4} = -\frac{1}{2}

. For Case 2,

m = -\frac{-4}{8} = \frac{1}{2}

.

The slope of a line with intercepts

(a, 0)

and

(0, b)

is given by

-\frac{b}{a}

.

Key Concept

Slope of a Line

Estimated Time:2m 0s

Question 2365Question →

In a triangle, the lengths of the sides are $x$ , $y$ , and $z$ , where $x$ , $y$ , and $z$ are integers such that $x < y < z$ . If $x = 7$ and the perimeter of the triangle is $32$ , what is the number of possible integer values for $z$ ?

Show answer & explanation

Answer: 3

Answer

There are exactly 3 possible integer values for the side length z.

The correct answer is 3. By expressing the second side as

y = 25 - z

and applying the ordering constraint

7 < 25 - z < z

, we determine that

12.5 < z < 18

. Applying the Triangle Inequality Theorem (

7 + y > z

) yields the restriction

z < 16

. Combining these conditions restricts the integer values of

z

to

\{13, 14, 15\}

, which counts to exactly 3 possible values.

Step-by-Step Solution

1

Express the side length

y

in terms of

z

.

y = 25 - z

The perimeter of the triangle is the sum of the side lengths:

x + y + z = 32

. Substituting

x = 7

gives

7 + y + z = 32

, which simplifies to

y = 25 - z

.

2

Apply the given inequality constraint

x < y < z

to find initial bounds for

z

.

12.5 < z < 18

Substituting

x = 7

and

y = 25 - z

into

x < y < z

yields

7 < 25 - z < z

. The left inequality

7 < 25 - z

simplifies to

z < 18

. The right inequality

25 - z < z

simplifies to

25 < 2z

, or

z > 12.5

.

3

Apply the Triangle Inequality Theorem to establish the final constraint on

z

.

z < 16

Since

z

is the longest side, the sum of the two shorter sides must be strictly greater than

z

:

x + y > z

. Substituting

x = 7

and

y = 25 - z

gives

7 + 25 - z > z

, which simplifies to

32 > 2z

, or

z < 16

.

4

Combine all constraints and count the valid integer values for

z

.

3 possible values (

13, 14, 15

)

Combining the bounds from the steps gives

12.5 < z < 16

. The integers satisfying this inequality are

13

,

14

, and

15

, which gives a total of 3 possible integer values.

Key Concept

Triangle Inequality Theorem and algebraic constraints on side lengths

Question 2366Question →

A shipping company charges a rate based on the weight of a package. The total cost $C$ , in dollars, to ship a package of weight $w$ pounds is given by the formula $C = \frac{5}{8}(w - 2) + 6.50$ for packages weighing more than $2$ pounds. If the shipping cost for a certain package is $\$14.00$ , what is the weight of the package, in pounds?

Show answer & explanation

Answer: 14

Answer

The weight of the package is 14 pounds.

Substituting

C = 14.00

into the formula gives

14.00 = \frac{5}{8}(w - 2) + 6.50

. Subtracting

6.50

from both sides yields

7.50 = \frac{5}{8}(w - 2)

. Multiplying both sides by the reciprocal

\frac{8}{5}

yields

12 = w - 2

. Finally, adding

2

to both sides gives the weight

w = 14

pounds.

Step-by-Step Solution

1

Substitute the total shipping cost into the formula.

14.00 = \frac{5}{8}(w - 2) + 6.50

Since the shipping cost

C

is given as

14.00

, we substitute this value into the formula to solve for the unknown weight

w

.

2

Subtract

6.50

from both sides of the equation.

7.50 = \frac{5}{8}(w - 2)

Subtracting

6.50

isolates the term containing the variable

w

on the right side of the equation.

3

Multiply both sides of the equation by the reciprocal of the fraction.

12 = w - 2

Multiplying by

\frac{8}{5}

eliminates the fractional coefficient of

\frac{5}{8}

on the right side.

4

Add

2

to both sides of the equation to solve for

w

.

w = 14

Adding

2

isolates the variable

w

, giving the final weight of the package.

Key Concept

Solving linear equations with fractional and decimal terms

Estimated Time:1m 30s

Question 2367Question →

In the standard $(x, y)$ coordinate plane, one diagonal of a square lies along the line with equation $y = 3x - 4$ . If one of the vertices of the square that does not lie on this diagonal is located at the point $(2, 8)$ , which of the following equations represents the line containing the other diagonal of the square?

3x - y = -2

3x + y = 14

x + 3y = 26

x - 3y = -22

x + 3y = 40

Show answer & explanation

Answer: $x + 3y = 26$

Answer

The equation of the line containing the other diagonal is

x + 3y = 26

.

The correct equation is

x + 3y = 26

. Since the diagonals of a square are perpendicular, their slopes must be negative reciprocals of each other. The given diagonal has a slope of

3

, meaning the other diagonal has a slope of

-\frac{1}{3}

. Since the vertex

(2, 8)

does not satisfy the equation of the first diagonal, it must lie on the second diagonal. Substituting this point into the slope-intercept or point-slope equation yields the standard form equation

x + 3y = 26

.

Step-by-Step Solution

1

Identify the slope of the given diagonal line.

The slope of the line

y = 3x - 4

is

3

.

The equation is in slope-intercept form

y = mx + b

, where

m

is the slope.

2

Determine the relationship between the two diagonals of a square.

The diagonals of a square are perpendicular to each other.

By geometric definition, the diagonals of any square intersect at right angles.

3

Calculate the slope of the perpendicular diagonal.

The slope of the perpendicular diagonal is

-\frac{1}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other, so the perpendicular slope is

-\frac{1}{3}

.

4

Identify the point that the perpendicular diagonal passes through.

The perpendicular diagonal passes through the vertex

(2, 8)

.

The vertex

(2, 8)

does not lie on the first diagonal because

8 \neq 3(2) - 4

. Since a square only has two diagonals, any vertex not on the first diagonal must lie on the second diagonal.

5

Find the equation of the perpendicular diagonal using point-slope form.

The equation is

x + 3y = 26

.

Using the point-slope form

y - y_1 = m(x - x_1)

with

m = -\frac{1}{3}

and point

(2, 8)

yields

y - 8 = -\frac{1}{3}(x - 2)

, which simplifies to

3(y - 8) = -(x - 2) \Rightarrow 3y - 24 = -x + 2 \Rightarrow x + 3y = 26

.

Key Concept

Diagonals of a square are perpendicular, and perpendicular lines have slopes that are negative reciprocals.

Question 2368Question →

A line segment in the standard $(x, y)$ coordinate plane has endpoints at $A(-1, 10)$ and $B(7, -2)$ . If point $M$ is the midpoint of this segment, what is the distance from $M$ to the origin $(0, 0)$ ?

Show answer & explanation

Answer: 5

Answer

The distance from the midpoint

M

to the origin is 5.

To find the distance from the midpoint

M

to the origin

(0,0)

, we first determine the coordinates of

M

by taking the average of the coordinates of

A(-1, 10)

and

B(7, -2)

. This results in

M\left(\frac{-1+7}{2}, \frac{10+(-2)}{2}\right) = (3, 4)

. Next, we apply the distance formula between

M(3,4)

and the origin

(0,0)

to get

\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9+16} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Calculate the coordinates of the midpoint

M

of the segment

AB

.

The midpoint is

M(3, 4)

.

The midpoint formula is

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

. Substituting the coordinates of

A(-1, 10)

and

B(7, -2)

gives the x-coordinate as

\frac{-1 + 7}{2} = 3

and the y-coordinate as

\frac{10 + (-2)}{2} = 4

.

2

Calculate the distance from the midpoint

M(3, 4)

to the origin

(0, 0)

.

The distance is 5.

The distance formula is

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Substituting the coordinates of

M(3, 4)

and the origin

(0, 0)

gives

d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

.

Key Concept

Using the midpoint formula to find the center point of a line segment, and then using the distance formula to find the length between that point and another specified coordinate.

Question 2369Question →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(\frac{1}{2}, \frac{2}{3})$ and $(\frac{3}{4}, \frac{5}{6})$ . What is the $y$ -intercept of this line?

\frac{1}{3}

\frac{1}{6}

-1

-\frac{1}{12}

-\frac{1}{7}

Show answer & explanation

Answer: $\frac{1}{3}$

Answer

The

y

-intercept of the line is

\frac{1}{3}

.

The correct answer is

\frac{1}{3}

. By calculating the slope of the line as

m = \frac{2}{3}

, and using the point-slope formula

y - y_1 = m(x - x_1)

with either given point, we can isolate

y

to get the equation

y = \frac{2}{3}x + \frac{1}{3}

, where the constant term represents the

y

-intercept.

Step-by-Step Solution

1

Calculate the slope (

m

) of the line passing through the points

(\frac{1}{2}, \frac{2}{3})

and

(\frac{3}{4}, \frac{5}{6})

using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

m = \frac{\frac{5}{6} - \frac{2}{3}}{\frac{3}{4} - \frac{1}{2}} = \frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3}

To find the equation of a line, we first need to determine its slope.

2

Use the point-slope form

y - y_1 = m(x - x_1)

with the point

(\frac{1}{2}, \frac{2}{3})

and slope

m = \frac{2}{3}

to write the equation of the line.

y - \frac{2}{3} = \frac{2}{3}(x - \frac{1}{2}) \Rightarrow y - \frac{2}{3} = \frac{2}{3}x - \frac{1}{3}

This allows us to write the linear equation in a form where we can isolate the

y

-intercept.

3

Isolate

y

to convert the equation into slope-intercept form

y = mx + b

, where

b

represents the

y

-intercept.

y = \frac{2}{3}x + \frac{1}{3}

The constant term in slope-intercept form represents the

y

-intercept.

Key Concept

Linear Equations and Graphing

Alternative Method

Instead of using the point-slope form, you can substitute one of the points directly into the slope-intercept form

y = mx + b

. Substituting

(\frac{1}{2}, \frac{2}{3})

and

m = \frac{2}{3}

yields

\frac{2}{3} = \frac{2}{3}(\frac{1}{2}) + b

, which simplifies to

\frac{2}{3} = \frac{1}{3} + b

. Solving for

b

gives

b = \frac{1}{3}

.

Estimated Time:1m 30s

Question 2370Question →

In a right triangle, the length of the side opposite the $60^\circ$ angle is $6\sqrt{3}$ centimeters. What is the length, in centimeters, of the hypotenuse of this triangle?

Show answer & explanation

Answer: 12

Answer

The length of the hypotenuse is

12

centimeters.

In a

30^\circ-60^\circ-90^\circ

special right triangle, the sides opposite the

30^\circ

angle, the

60^\circ

angle, and the

90^\circ

(hypotenuse) angle are in the ratio

x : x\sqrt{3} : 2x

. Given that the side opposite the

60^\circ

angle is

6\sqrt{3}

centimeters, we have

x\sqrt{3} = 6\sqrt{3}

, which means

x = 6

. The hypotenuse is

2x = 2(6) = 12

centimeters.

Step-by-Step Solution

1

Determine the type of special right triangle.

A

30^\circ-60^\circ-90^\circ

right triangle.

Since the triangle is a right triangle and has a

60^\circ

angle, the remaining angle must be

180^\circ - 90^\circ - 60^\circ = 30^\circ

.

2

Set up the relation for the side lengths using the ratio of a

30^\circ-60^\circ-90^\circ

triangle.

The side opposite the

60^\circ

angle is

x\sqrt{3}

centimeters, where

x

is the length of the side opposite the

30^\circ

angle.

In any

30^\circ-60^\circ-90^\circ

triangle, the side lengths are in the ratio

1 : \sqrt{3} : 2

.

3

Solve for the base variable

x

.

x = 6

We are given that the side opposite the

60^\circ

angle is

6\sqrt{3}

centimeters, so

x\sqrt{3} = 6\sqrt{3}

.

4

Calculate the length of the hypotenuse.

The hypotenuse is

2x = 2(6) = 12

centimeters.

The hypotenuse of a

30^\circ-60^\circ-90^\circ

triangle is twice the length of the shorter leg, which is

2x

.

Key Concept

Using the side length ratios of a

30^\circ-60^\circ-90^\circ

special right triangle to find missing lengths.

Question 2371Question →

A triangle has two sides of length 5 and 12. The third side has a length of $x$ , where $x$ is an integer. If the perimeter of the triangle is a multiple of 5, what is the sum of all possible values of $x$ ?

13

25

21

24

39

Show answer & explanation

Answer: 21

Answer

The sum of all possible values of the third side length is 21.

To find the sum of all possible values of

x

, we first apply the Triangle Inequality Theorem. For a triangle with side lengths 5, 12, and

x

, the third side must satisfy

12 - 5 < x < 12 + 5

, which simplifies to

7 < x < 17

. The perimeter

P

of the triangle is given by

P = 5 + 12 + x = 17 + x

. Given that

7 < x < 17

, the perimeter must be between

17 + 7 = 24

and

17 + 17 = 34

. The only multiples of 5 within this range are 25 and 30. Setting the perimeter equal to these values gives

17 + x = 25 \implies x = 8

, and

17 + x = 30 \implies x = 13

. Both values are integers and satisfy the triangle inequality. The sum of these values is

8 + 13 = 21

.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to find the range of possible values for the third side,

x

.

12 - 5 < x < 12 + 5

, which simplifies to

7 < x < 17

.

The length of any side of a triangle must be strictly greater than the difference between the other two sides and strictly less than their sum.

2

Determine the expression for the perimeter of the triangle and find the bounds for the perimeter.

Perimeter

P = 5 + 12 + x = 17 + x

. Since

7 < x < 17

, the perimeter must satisfy

17 + 7 < P < 17 + 17

, which means

24 < P < 34

.

The perimeter of a triangle is the sum of its three side lengths.

3

Identify which values of the perimeter in this range are multiples of 5, and find the corresponding values of

x

.

The multiples of 5 between 24 and 34 are 25 and 30. If

P = 25

, then

17 + x = 25 \implies x = 8

. If

P = 30

, then

17 + x = 30 \implies x = 13

. Both

x = 8

and

x = 13

are integers that satisfy the initial inequality.

We must find the integer values of

x

that make the perimeter a multiple of 5.

4

Calculate the sum of all possible values of

x

.

8 + 13 = 21

.

The question asks for the sum of all valid integer values of

x

.

Key Concept

Triangle Inequality Theorem and Perimeter Calculations

Estimated Time:1m 30s

Question 2372Question →

On a coordinate grid, a circle is drawn such that its center is at the point $(4, -2)$ . If the circle is tangent to the $y$ -axis, what is the equation of this circle?

(x + 4)^2 + (y - 2)^2 = 16

(x - 4)^2 + (y + 2)^2 = 4

(x - 4)^2 + (y + 2)^2 = 16

(x + 4)^2 + (y - 2)^2 = 4

(x - 2)^2 + (y + 4)^2 = 16

Show answer & explanation

Answer: $(x - 4)^2 + (y + 2)^2 = 16$

Answer

(x - 4)^2 + (y + 2)^2 = 16

The correct equation is

(x - 4)^2 + (y + 2)^2 = 16

. A circle with center

(h, k)

and radius

r

has the standard equation

(x - h)^2 + (y - k)^2 = r^2

. Substituting

(4, -2)

for the center gives

(x - 4)^2 + (y + 2)^2 = r^2

. Since the circle is tangent to the

y

-axis, the radius is the horizontal distance from the center to the line

x = 0

, which is

4

units. Squaring the radius gives

r^2 = 16

.

Step-by-Step Solution

1

Recall the standard form of the equation of a circle.

The standard equation of a circle is

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center and

r

is the radius.

This sets up the formula needed to write the circle's equation.

2

Substitute the center coordinates

(h, k) = (4, -2)

into the formula.

(x - 4)^2 + (y - (-2))^2 = r^2

, which simplifies to

(x - 4)^2 + (y + 2)^2 = r^2

.

To represent the translation of the circle's center from the origin.

3

Determine the radius of the circle using the given tangency condition.

Since the circle is tangent to the

y

-axis, the radius is the horizontal distance from the center

(4, -2)

to the

y

-axis (the line

x = 0

). This distance is

|4 - 0| = 4

units, so

r = 4

.

To find the radius from the geometric properties of the circle.

4

Square the radius and write the complete equation.

r^2 = 4^2 = 16

. The completed equation is

(x - 4)^2 + (y + 2)^2 = 16

.

To complete the standard equation form.

Key Concept

Standard form of a circle's equation and determining its radius from a tangency condition

Question 2373Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ and the measure of angle $A$ is $45^\circ$ . If the length of leg $AB$ is $8$ inches, what is the length, in inches, of the hypotenuse $AC$ ?

8

4\sqrt{2}

8\sqrt{2}

8\sqrt{3}

16

Show answer & explanation

Answer: $8\sqrt{2}$

Answer

The length of the hypotenuse is

8\sqrt{2}

inches.

In right triangle

ABC

, the angle measures are

45^\circ

,

45^\circ

, and

90^\circ

. The lengths of the sides of a

45^\circ

-

45^\circ

-

90^\circ

triangle are in the ratio

1 : 1 : \sqrt{2}

. Since the leg is

8

inches, the hypotenuse is

8\sqrt{2}

inches.

Step-by-Step Solution

1

Identify the type of right triangle.

Since angle

B = 90^\circ

and angle

A = 45^\circ

, angle

C

must also be

45^\circ

. This is a

45^\circ

-

45^\circ

-

90^\circ

special right triangle.

The sum of the angles in a triangle is always

180^\circ

.

2

Recall the ratio of the side lengths of a

45^\circ

-

45^\circ

-

90^\circ

triangle.

The ratio of the sides opposite the angles

45^\circ : 45^\circ : 90^\circ

is

1 : 1 : \sqrt{2}

. Thus, the hypotenuse is equal to

\text{leg} \times \sqrt{2}

.

This is a standard geometric property of isosceles right triangles.

3

Calculate the length of the hypotenuse.

Multiply the leg length of

8

inches by

\sqrt{2}

to get

8\sqrt{2}

inches.

The leg adjacent to the

45^\circ

angle is given as

8

inches.

Key Concept

Hypotenuse of a

45^\circ

-

45^\circ

-

90^\circ

special right triangle

Alternative Method

Alternatively, use the Pythagorean theorem:

AB^2 + BC^2 = AC^2

. Since it is an isosceles right triangle,

BC = AB = 8

. Thus,

8^2 + 8^2 = AC^2 \implies 64 + 64 = AC^2 \implies AC = \sqrt{128} = 8\sqrt{2}

.

Estimated Time:45s

Question 2374Question →

A shipping company charges a flat fee of $15$ dollars plus $2.50$ dollars per pound for the first $10$ pounds of a package's weight. For any weight exceeding $10$ pounds, the rate is $4.00$ dollars per pound. If the total shipping charge for a package was $59.00$ dollars, what was the total weight of the package, in pounds?

11.0

11.6

14.75

17.6

18.5

Show answer & explanation

Answer: $14.75$

Answer

The package's total weight was 14.75 pounds.

The correct answer of 14.75 pounds is found by setting up the equation representing the total shipping cost. First, calculate the cost for the first 10 pounds, which is the flat fee of 15.00 dollars plus 2.50 dollars per pound for 10 pounds: 15 + 2.50(10) = 40.00 dollars. Since the total charge of 59.00 dollars is greater than 40.00 dollars, the package must weigh more than 10 pounds. Let w be the total weight of the package. The remaining weight exceeding 10 pounds is w - 10, which is charged at 4.00 dollars per pound. Setting up the equation: 40 + 4(w - 10) = 59. Solving for w gives: 4(w - 10) = 19, which simplifies to w - 10 = 4.75, so w = 14.75.

Step-by-Step Solution

1

Calculate the cost for a package weighing exactly 10 pounds.

The cost is 15 + 2.50 * 10 = 40.00 dollars.

To determine whether the package exceeds 10 pounds by comparing it to the total charge of 59.00 dollars.

2

Set up an equation for the total cost where the weight w exceeds 10 pounds.

40 + 4.00 * (w - 10) = 59.00

To express the cost of the first 10 pounds plus the cost of the excess weight at 4.00 dollars per pound.

3

Solve the equation for the total weight w.

4 * (w - 10) = 19 -> w - 10 = 4.75 -> w = 14.75

To isolate and find the value of the variable representing the package weight.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 2375Question →

The first term of an arithmetic sequence is $\frac{1}{2}$ , and the third term of the sequence is $\frac{5}{6}$ . What is the sum of the first 6 terms of this sequence?

\frac{9}{4}

3

5

\frac{11}{2}

\frac{21}{2}

Show answer & explanation

Answer: $\frac{11}{2}$

Answer

The sum of the first 6 terms is

\frac{11}{2}

.

To find the sum of the first 6 terms, we first determine the common difference

d

from the given terms:

a_3 = a_1 + 2d \Rightarrow \frac{5}{6} = \frac{1}{2} + 2d \Rightarrow 2d = \frac{1}{3} \Rightarrow d = \frac{1}{6}

. Then, we apply the arithmetic series sum formula:

S_6 = \frac{6}{2}[2(\frac{1}{2}) + 5(\frac{1}{6})] = 3[1 + \frac{5}{6}] = 3(\frac{11}{6}) = \frac{11}{2}

.

Step-by-Step Solution

1

Find the common difference

d

using the formula for the

n

-th term of an arithmetic sequence

a_n = a_1 + (n - 1)d

with the given values

a_1 = \frac{1}{2}

and

a_3 = \frac{5}{6}

.

2d = \frac{5}{6} - \frac{1}{2} \Rightarrow 2d = \frac{1}{3} \Rightarrow d = \frac{1}{6}

We need to find the common difference to determine the subsequent terms and calculate the sum of the sequence.

2

Use the sum formula

S_n = \frac{n}{2}[2a_1 + (n - 1)d]

with

n = 6

,

a_1 = \frac{1}{2}

, and

d = \frac{1}{6}

to find the sum of the first 6 terms.

S_6 = \frac{6}{2}[2(\frac{1}{2}) + (6 - 1)(\frac{1}{6})] = 3[1 + \frac{5}{6}] = 3(\frac{11}{6}) = \frac{11}{2}

This formula sums the first 6 terms of the arithmetic sequence directly.

Key Concept

Sum of the first

n

terms of an arithmetic sequence using fractional terms

Alternative Method

Alternatively, you can list the first 6 terms of the sequence and add them directly:

\frac{3}{6}, \frac{4}{6}, \frac{5}{6}, \frac{6}{6}, \frac{7}{6}, \frac{8}{6}

. Adding these gives

\frac{33}{6} = \frac{11}{2}

.

Estimated Time:1m 30s

Question 2376Question →

A craft cider company produces two specialty blends. The production cost, in dollars per gallon, of the premium blend is represented by the expression $\frac{3}{4}(d - 8)$ , where $d$ is the wholesale cost, in dollars, of a bushel of apples. The production cost, in dollars per gallon, of the dry blend is represented by the expression $0.2(2d + 5)$ . If the production cost per gallon is the same for both blends, what is this production cost, in dollars per gallon?

1.2

1.8

9

20

45

Show answer & explanation

Answer: 9

Answer

The production cost is 9 dollars per gallon.

The correct answer is the value obtained by setting the two cost expressions equal, solving for

d = 20

, and then substituting

d = 20

back into either expression to find the cost of

9

dollars per gallon.

Step-by-Step Solution

1

Set the two production cost expressions equal to each other.

\frac{3}{4}(d - 8) = 0.2(2d + 5)

The problem states that the production cost per gallon is the same for both blends, so their algebraic representations must be equal.

2

Express the decimal as a fraction and solve for

d

.

15(d - 8) = 4(2d + 5) \implies 15d - 120 = 8d + 20 \implies 7d = 140 \implies d = 20

Converting

0.2

to

\frac{1}{5}

allows us to clear fractions by multiplying both sides of the equation by the least common multiple of the denominators, which is

20

. We then distribute, isolate the variable terms, and solve for

d

.

3

Substitute the value of

d

back into either of the original cost expressions to find the production cost.

\text{Cost} = \frac{3}{4}(20 - 8) = \frac{3}{4}(12) = 9

The question asks for the production cost per gallon, not the value of the variable

d

. Substituting

d = 20

into the premium blend cost expression gives the final cost.

Key Concept

Solving linear equations involving fractions and decimals, and evaluating expressions using the solved variable value.

Alternative Method

Convert all numbers to decimals. The premium blend cost is

0.75(d - 8) = 0.75d - 6

. The dry blend cost is

0.2(2d + 5) = 0.4d + 1

. Equating them gives

0.75d - 6 = 0.4d + 1 \implies 0.35d = 7 \implies d = 20

. Then find the cost:

0.75(20) - 6 = 15 - 6 = 9

dollars.

Estimated Time:1m 30s

Question 2377Question →

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $A(k, 4)$ , $B(-1, 2)$ , and $C(3, -6)$ . The altitude from vertex $A$ to side $BC$ intersects the $y$ -axis at $(0, -2)$ . What is the value of $k$ ?

-12

3

4

12

24

Show answer & explanation

Answer: 12

Answer

12

The slope of side

BC

is calculated as

m_{BC} = \frac{-6 - 2}{3 - (-1)} = -2

. Since the altitude from vertex

A

is perpendicular to side

BC

, its slope must be the negative reciprocal of

-2

, which is

\frac{1}{2}

. The equation of the line containing this altitude, with a given

y

-intercept of

(0, -2)

, is

y = \frac{1}{2}x - 2

. Substituting the coordinates of vertex

A(k, 4)

into the equation gives

4 = \frac{1}{2}k - 2

. Solving for

k

yields

k = 12

.

Step-by-Step Solution

1

Calculate the slope of side

BC

using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

m_{BC} = \frac{-6 - 2}{3 - (-1)} = \frac{-8}{4} = -2

The altitude is perpendicular to the side

BC

, so we first need the slope of

BC

.

2

Find the slope of the altitude by taking the negative reciprocal of the slope of

BC

.

m_{\text{alt}} = -\frac{1}{m_{BC}} = -\frac{1}{-2} = \frac{1}{2}

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Write the equation of the line containing the altitude using the slope-intercept form

y = mx + b

with the given

y

-intercept of

(0, -2)

.

y = \frac{1}{2}x - 2

The line has a slope of

\frac{1}{2}

and crosses the

y

-axis at

-2

.

4

Substitute the coordinates of vertex

A(k, 4)

into the equation and solve for

k

.

4 = \frac{1}{2}k - 2 \Rightarrow 6 = \frac{1}{2}k \Rightarrow k = 12

Vertex

A

lies on the altitude line, so its coordinates must satisfy the line's equation.

Key Concept

The slope of a line perpendicular to a given line is the negative reciprocal of the given line's slope.

Alternative Method

Alternatively, we can use the vector dot product. The vector representing side

BC

is

\vec{BC} = (3 - (-1), -6 - 2) = (4, -8)

. The vector from the

y

-intercept to vertex

A

is

\vec{v} = (k - 0, 4 - (-2)) = (k, 6)

. Since the altitude is perpendicular to

BC

, the dot product of these two vectors must equal zero:

(4)(k) + (-8)(6) = 0 \Rightarrow 4k - 48 = 0 \Rightarrow 4k = 48 \Rightarrow k = 12

.

Estimated Time:2m 0s

Question 2378Question →

A circle in the standard $(x, y)$ coordinate plane passes through the points $(0, 0)$ , $(8, 0)$ , and $(2, 6)$ . What are the coordinates of the center of this circle?

(3, 3)

(4, 2)

(4, 3)

(5, 3)

(2, 4)

Show answer & explanation

Answer: $(4, 2)$

Answer

The center of the circle is

(4, 2)

The correct answer is

(4, 2)

. A circle's center

(h, k)

is equidistant from all points on the circle. By setting up the distance formula from the center

(h, k)

to the points

(0,0)

,

(8,0)

, and

(2,6)

, we obtain the equations

h^2 + k^2 = (h-8)^2 + k^2

and

h^2 + k^2 = (h-2)^2 + (k-6)^2

. Solving the first equation yields

16h = 64

, which simplifies to

h = 4

. Substituting

h = 4

into the second equation yields

16 + k^2 = 4 + (k-6)^2

. Expanding and simplifying gives

16 = 40 - 12k

, which simplifies to

12k = 24

, or

k = 2

. Therefore, the center of the circle is

(4, 2)

.

Step-by-Step Solution

1

Set up the distance formula equations for the center of the circle,

(h, k)

, which must be equidistant from the three points on the circle:

(0, 0)

,

(8, 0)

, and

(2, 6)

.

The distances from

(h, k)

to the points are

\sqrt{h^2 + k^2}

,

\sqrt{(h-8)^2 + k^2}

, and

\sqrt{(h-2)^2 + (k-6)^2}

. Since they are radii, we equate them:

h^2 + k^2 = (h-8)^2 + k^2 = (h-2)^2 + (k-6)^2

.

The distance from the center of a circle to any point on its circumference is equal to its radius.

2

Equate the distance to

(0,0)

and the distance to

(8,0)

, then solve for

h

.

h^2 + k^2 = (h-8)^2 + k^2 \implies h^2 = h^2 - 16h + 64 \implies 16h = 64 \implies h = 4

.

Since both points lie on the x-axis, the x-coordinate of the center must lie on the perpendicular bisector of the segment connecting them, which is

x = 4

.

3

Substitute

h = 4

into the equation equating the distance to

(0,0)

and

(2,6)

to solve for

k

.

4^2 + k^2 = (4-2)^2 + (k-6)^2 \implies 16 + k^2 = 4 + k^2 - 12k + 36 \implies 16 = 40 - 12k \implies 12k = 24 \implies k = 2

.

Substituting the known coordinate allows us to solve the remaining single-variable equation for the

y

-coordinate of the center.

4

Combine the coordinates to state the final center of the circle.

The center is

(4, 2)

.

Both coordinates

(h, k)

have been uniquely determined.

Key Concept

Circumcenter of a triangle / Center of a circle passing through three points on a coordinate plane

Estimated Time:2m 0s

Question 2379Question →

In the standard $(x, y)$ coordinate plane, if the three distinct points $(k, 4)$ , $(4, k)$ , and $(5, 6)$ are collinear, then there are exactly two possible values for the slope of the line passing through these points.

True

False

Show answer & explanation

Answer: False

Answer

The statement is false because the requirement that the three points must be distinct eliminates one of the algebraic solutions for

k

, resulting in only one possible slope for the line.

The statement is false because equating the slopes between the points leads to the quadratic equation

k^2 - 11k + 28 = 0

, which has the solutions

k = 4

and

k = 7

. However, substituting

k = 4

back into the coordinates results in the points

(4, 4)

and

(4, 4)

, which are not distinct. Thus, only

k = 7

is a valid solution, which yields exactly one line with a slope of

-1

.

Step-by-Step Solution

1

Set up the collinearity condition using the slope formula.

The slope of the segment connecting

(4, k)

and

(5, 6)

is

m_1 = \frac{6 - k}{5 - 4} = 6 - k

. The slope of the segment connecting

(k, 4)

and

(5, 6)

is

m_2 = \frac{6 - 4}{5 - k} = \frac{2}{5 - k}

. Equating the two slopes gives

6 - k = \frac{2}{5 - k}

.

For three points to be collinear, the slope between any two pairs of points must be equal.

2

Solve the rational equation for

k

.

Multiplying both sides by

5 - k

yields

(6 - k)(5 - k) = 2

, which expands to

30 - 11k + k^2 = 2

. Rearranging into standard quadratic form gives

k^2 - 11k + 28 = 0

. Factoring the quadratic yields

(k - 4)(k - 7) = 0

, giving the solutions

k = 4

or

k = 7

.

Finding the roots of the quadratic equation identifies the algebraic candidates for

k

.

3

Verify if the points are distinct for each candidate value of

k

.

If

k = 4

, the first two points are

(4, 4)

and

(4, 4)

, which are not distinct. If

k = 7

, the three points are

(7, 4)

,

(4, 7)

, and

(5, 6)

, which are distinct.

The problem specifies that the three points must be distinct, so any solution violating this condition must be discarded.

4

Determine the number of possible slopes.

For the only valid value

k = 7

, the slope of the line is

m = 6 - 7 = -1

. There is only one possible slope.

Comparing the number of valid slopes to the statement shows that the statement is false.

Key Concept

Collinearity of points in the coordinate plane and constraints on slope calculation.

Question 2380Question →

A straight line graphed on a coordinate grid has a slope of $1.25$ and passes through the point $(4, 11)$ . If the point $(k, -4)$ also lies on this line, what is the value of $k$ ?

Show answer & explanation

Answer: -8

Answer

The value of

k

is

-8

.

The correct answer is

-8

. By using the point-slope formula of a line, we determine the equation to be

y - 11 = 1.25(x - 4)

. Substituting the coordinates of the point

(k, -4)

yields the equation

-4 - 11 = 1.25(k - 4)

. Solving this equation results in

k = -8

.

Step-by-Step Solution

1

Use the point-slope formula of a line,

y - y_1 = m(x - x_1)

, to write the equation of the line.

y - 11 = 1.25(x - 4)

We are given the slope

m = 1.25

and a point

(4, 11)

on the line.

2

Substitute the coordinates of the second point,

(k, -4)

, into the line's equation.

-4 - 11 = 1.25(k - 4)

Since the point

(k, -4)

lies on the line, its coordinates must satisfy the line's equation.

3

Solve the equation for

k

.

k = -8

Simplify the left side to get

-15 = 1.25(k - 4)

, divide both sides by

1.25

to get

-12 = k - 4

, and add

4

to both sides to find

k = -8

.

Key Concept

Writing and solving linear equations using point-slope form.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer