All practice questions

Question 281Question →

A circle in the standard $(x,y)$ coordinate plane is defined by the equation $(x - 8)^2 + (y + 5)^2 = 121$ . What is the diameter of this circle?

Show answer & explanation

Answer: 22

Answer

The diameter of the circle is 22.

By comparing the given equation

(x - 8)^2 + (y + 5)^2 = 121

to the standard circle equation

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

, we find that

r^2 = 121

. Taking the square root of

121

gives a radius of

r = 11

. Since the diameter of a circle is twice the radius, the diameter is

2 \times 11 = 22

.

Step-by-Step Solution

1

Identify the relationship between the circle's equation and its radius squared.

r^2 = 121

The standard equation of a circle is

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

, meaning the constant on the right side of the equation represents the square of the radius.

2

Determine the radius of the circle.

r = 11

Taking the square root of

121

yields the radius of the circle, which must be a positive length.

3

Calculate the diameter of the circle.

d = 22

The diameter of a circle is defined as twice the length of its radius (

d = 2r

).

Key Concept

The standard equation of a circle is

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

, where

r

is the radius and the diameter is

2r

.

Question 282Question →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(2, 5)$ and $(6, k)$ . If the $y$ -intercept of this line is $-1$ , what is the value of $k$ ?

Show answer & explanation

Answer: 17

Answer

The value of

k

is 17.

The correct answer is 17. The y-intercept of

-1

gives the point

(0, -1)

. The slope between

(0, -1)

and

(2, 5)

is

\frac{5 - (-1)}{2 - 0} = 3

. The equation of the line is

y = 3x - 1

. Substituting the point

(6, k)

into the equation gives

k = 3(6) - 1 = 17

.

Step-by-Step Solution

1

Identify the coordinates of the y-intercept.

The point is

(0, -1)

.

The y-intercept is the point where the line crosses the y-axis, which occurs when the x-coordinate is 0.

2

Calculate the slope (

m

) of the line using the points

(0, -1)

and

(2, 5)

.

m = 3

Using the slope formula

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

, we find

m = \frac{5 - (-1)}{2 - 0} = \frac{6}{2} = 3

.

3

Write the equation of the line.

y = 3x - 1

Using the slope-intercept form

y = mx + b

, where the slope

m = 3

and the y-intercept

b = -1

.

4

Substitute the point

(6, k)

into the line equation to solve for

k

.

k = 17

Since the point

(6, k)

lies on the line, substituting

x = 6

and

y = k

yields

k = 3(6) - 1 = 18 - 1 = 17

.

Key Concept

Linear Equations and Graphing

Alternative Method

Since the slope of a straight line is constant, the slope between

(0, -1)

and

(2, 5)

must equal the slope between

(2, 5)

and

(6, k)

. Setting the slopes equal:

\frac{5 - (-1)}{2 - 0} = \frac{k - 5}{6 - 2} \implies 3 = \frac{k - 5}{4} \implies 12 = k - 5 \implies k = 17

.

Estimated Time:1m 0s

Question 283Question →

For all positive real numbers $w$ , the expression $\left(\frac{w^4 \cdot \sqrt{w}}{w^{-1.5}}\right)^{2/3}$ is equivalent to $w^k$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Applying the exponent rules in sequence: first, rewrite the square root as a fractional exponent to get

w^{0.5}

. Next, multiply the terms in the numerator by adding their exponents:

4 + 0.5 = 4.5

. Then, divide the numerator by the denominator by subtracting the denominator's exponent from the numerator's exponent:

4.5 - (-1.5) = 6

. Finally, raise this result to the

2/3

power by multiplying the exponents:

6 \times (2/3) = 4

. This yields

w^4

, so the constant exponent is

4

.

Step-by-Step Solution

1

Convert the radical expression to an exponential expression.

\sqrt{w} = w^{0.5}

Converting all terms to base

w

with decimal or fractional exponents makes it easier to apply exponent properties.

2

Apply the product rule of exponents to the numerator.

w^4 \cdot w^{0.5} = w^{4 + 0.5} = w^{4.5}

When multiplying exponential terms with the same base, add their exponents:

w^a \cdot w^b = w^{a+b}

.

3

Apply the quotient rule of exponents to the fraction.

\frac{w^{4.5}}{w^{-1.5}} = w^{4.5 - (-1.5)} = w^6

When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator:

\frac{w^a}{w^b} = w^{a-b}

.

4

Apply the power rule of exponents to the simplified term.

(w^6)^{2/3} = w^{6 \cdot \frac{2}{3}} = w^4

When raising a power to another power, multiply the exponents:

(w^a)^b = w^{a \cdot b}

.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 284Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ is perpendicular to the line represented by the equation $3x - 2y = 6$ . Line $L_2$ is parallel to $L_1$ and passes through the midpoint of the line segment with endpoints $(-1, 5)$ and $(3, -1)$ . If the equation of $L_2$ is written in the form $Ax + By = 8$ , where $A$ and $B$ are integers, what is the value of $A + B$ ?

Show answer & explanation

Answer: 5

Answer

The value of

A + B

is 5.

The given line

3x - 2y = 6

has a slope of

\frac{3}{2}

. A perpendicular line

L_1

has a slope of

-\frac{2}{3}

. A line

L_2

parallel to

L_1

also has a slope of

-\frac{2}{3}

. The midpoint of the segment with endpoints

(-1, 5)

and

(3, -1)

is calculated as

\left(\frac{-1+3}{2}, \frac{5-1}{2}\right) = (1, 2)

. Using the point-slope form with the slope

-\frac{2}{3}

and point

(1, 2)

yields

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

, which simplifies to

2x + 3y = 8

. Thus,

A = 2

,

B = 3

, and

A + B = 5

.

Step-by-Step Solution

1

Find the slope of the given line

3x - 2y = 6

.

Slope is

\frac{3}{2}

Convert the equation to slope-intercept form

y = mx + b

to find the slope.

2

Determine the slope of line

L_1

.

Slope is

-\frac{2}{3}

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

3

Determine the slope of line

L_2

.

Slope is

-\frac{2}{3}

Parallel lines have the same slope.

4

Calculate the midpoint of the segment with endpoints

(-1, 5)

and

(3, -1)

.

Midpoint is

(1, 2)

Use the midpoint formula:

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

.

5

Write the equation of line

L_2

and convert it to standard form.

Equation is

2x + 3y = 8

Use the point-slope form with slope

-\frac{2}{3}

and midpoint

(1, 2)

, then rearrange to match the target constant 8 on the right-hand side.

6

Identify

A

and

B

, and compute their sum.

A = 2

,

B = 3

, and

A + B = 5

Match the coefficients of

2x + 3y = 8

to

Ax + By = 8

and add them.

Key Concept

Determining equations of parallel and perpendicular lines on the coordinate plane using slopes and midpoints.

Question 285Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $(x - 2)^2 + y^2 = 20$ , and a line is defined by the equation $y = 2x - 4$ . If the circle and the line intersect at two points, $(x_1, y_1)$ and $(x_2, y_2)$ , what is the value of $x_1 + x_2 + y_1 + y_2$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x_1 + x_2 + y_1 + y_2

is 4.

Substituting the equation of the line into the circle's equation gives

(x-2)^2 + (2x-4)^2 = 20

. Since

(2x-4) = 2(x-2)

, we can write this as

(x-2)^2 + 4(x-2)^2 = 20

, which simplifies to

5(x-2)^2 = 20

, and further to

(x-2)^2 = 4

. Solving for

x

yields

x - 2 = 2

or

x - 2 = -2

, so

x = 4

or

x = 0

. Substituting these values into the linear equation gives the points

(4, 4)

and

(0, -4)

. The sum of all coordinates is

4 + 0 + 4 + (-4) = 4

.

Step-by-Step Solution

1

Substitute the linear equation into the circle equation.

(x - 2)^2 + (2x - 4)^2 = 20

To find the coordinates of the intersection points by reducing the system to a single variable equation.

2

Simplify the equation using algebraic substitution.

5(x - 2)^2 = 20

Factoring 2 out of the term

(2x-4)

yields

2(x-2)

. Squaring it results in

4(x-2)^2

. Combining this with the first term

(x-2)^2

simplifies the expression.

3

Solve for the two possible x-coordinates.

x = 0

and

x = 4

Dividing both sides by 5 yields

(x-2)^2 = 4

, which means

x-2 = \pm 2

.

4

Substitute the x-coordinates back into the linear equation to find the corresponding y-coordinates.

For

x = 0

,

y = -4

, giving the point

(0, -4)

. For

x = 4

,

y = 4

, giving the point

(4, 4)

.

To determine the coordinates of both intersection points.

5

Calculate the sum of all coordinates:

x_1 + x_2 + y_1 + y_2

.

0 + 4 + (-4) + 4 = 4

To find the final value requested by the question.

Key Concept

Solving systems of linear and quadratic (circular) equations by substitution and factoring.

Question 286Question →

In the standard $(x, y)$ coordinate plane, a rectangle has vertices at $(1, 2)$ , $(7, 2)$ , $(7, 10)$ , and $(1, 10)$ . What is the length of a diagonal of this rectangle?

Show answer & explanation

Answer: 10

Answer

The length of a diagonal of the rectangle is 10.

The width of the rectangle is

7 - 1 = 6

, and the height is

10 - 2 = 8

. The diagonal forms the hypotenuse of a right triangle with legs of 6 and 8. By the Pythagorean theorem, the length of the diagonal is

\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

.

Step-by-Step Solution

1

Calculate the horizontal width of the rectangle.

Width =

7 - 1 = 6

The horizontal distance is found by subtracting the x-coordinates of the horizontal vertices.

2

Calculate the vertical height of the rectangle.

Height =

10 - 2 = 8

The vertical distance is found by subtracting the y-coordinates of the vertical vertices.

3

Apply the Pythagorean theorem to find the diagonal length.

Diagonal length =

\sqrt{6^2 + 8^2} = 10

The diagonal of a rectangle forms the hypotenuse of a right triangle with legs equal to the width and the height.

Key Concept

Using coordinate differences to find dimensions of a figure and using the Pythagorean theorem to find its diagonal length.

Question 287Question →

A food truck selling gourmet grilled cheese sandwiches has a daily fixed operating cost of $120$ . Each sandwich costs $2.50$ to make and is sold for $6.50$ . What is the minimum number of sandwiches the food truck must sell in one day to make a net profit of at least $180$ ?

Show answer & explanation

Answer: 75

Answer

The food truck must sell a minimum of 75 sandwiches to make a net profit of at least $180.

Representing the number of sandwiches sold as

x

, the total revenue is

6.50x

and the total cost is

120 + 2.50x

. The net profit is the difference between revenue and cost:

6.50x - (120 + 2.50x)

, which simplifies to

4x - 120

. Setting up the inequality for a profit of at least

180

gives

4x - 120 \geq 180

. Solving for

x

gives

4x \geq 300

, which simplifies to

x \geq 75

. Therefore, the minimum number of sandwiches that must be sold is

75

.

Step-by-Step Solution

1

Define the variable and write expressions for revenue and cost.

Let

x

represent the number of sandwiches sold. Total Revenue =

6.50x

and Total Cost =

120 + 2.50x

.

Defining the variable and translating the verbal descriptions of revenue and cost into algebraic expressions is necessary to model the profit.

2

Formulate the net profit expression.

Net Profit = Total Revenue - Total Cost =

6.50x - (120 + 2.50x) = 4x - 120

.

Net profit is calculated by subtracting all fixed and variable costs from the total revenue.

3

Set up and solve the linear inequality.

4x - 120 \geq 180 \Rightarrow 4x \geq 300 \Rightarrow x \geq 75

.

To find the minimum number of sandwiches needed to reach a target profit of at least

180

, we solve the inequality

4x - 120 \geq 180

for

x

.

Key Concept

Translating a real-world scenario into a linear inequality and solving for the unknown variable.

Question 288Question →

A line in the standard $(x, y)$ coordinate plane passes through the points $(3, -2)$ and $(7, 6)$ . What is the slope of this line?

Show answer & explanation

Answer: 2

Answer

The slope of the line is

2

.

The correct answer is

2

. The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

. Substituting

(3, -2)

and

(7, 6)

gives

m = \frac{6 - (-2)}{7 - 3} = \frac{8}{4} = 2

.

Step-by-Step Solution

1

Identify the coordinates of the two points on the line.

(x_1, y_1) = (3, -2)

and

(x_2, y_2) = (7, 6)

The coordinates are needed to apply the slope formula.

2

Apply the slope formula

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

.

m = \frac{6 - (-2)}{7 - 3}

The slope is the ratio of the vertical change (rise) to the horizontal change (run).

3

Simplify the numerator and the denominator, then divide.

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

Subtracting a negative value is equivalent to addition, which yields the final slope of

2

.

Key Concept

Slope formula

Question 289Question →

A business analyst uses the linear equation $\frac{2}{5}(3p - 10) - 1.2 = 0.4(p + 5)$ to estimate the equilibrium price $p$ , in dollars, of a new product. What is the equilibrium price, in dollars, of the product?

Show answer & explanation

Answer: 9

Answer

The equilibrium price of the product is

9

dollars.

Converting the fraction to a decimal gives

0.4(3p - 10) - 1.2 = 0.4(p + 5)

. Dividing both sides by

0.4

results in

3p - 10 - 3 = p + 5

, which simplifies to

3p - 13 = p + 5

. Subtracting

p

and adding

13

to both sides gives

2p = 18

, and dividing by

2

results in

9

.

Step-by-Step Solution

1

Convert the fraction to decimal form

0.4(3p - 10) - 1.2 = 0.4(p + 5)

Converting

\frac{2}{5}

to

0.4

makes all terms decimals, simplifying further operations.

2

Divide both sides of the equation by

0.4

(3p - 10) - 3 = p + 5

Since

0.4

is a common factor and

1.2 / 0.4 = 3

, dividing both sides by

0.4

simplifies the coefficients.

3

Simplify the left side of the equation

3p - 13 = p + 5

Combine the constant terms

-10

and

-3

to simplify the expression.

4

Isolate the variable terms on one side

2p = 18

Subtract

p

from both sides and add

13

to both sides.

5

Solve for

p

p = 9

Divide both sides by

2

to find the final value.

Key Concept

Solving Linear Equations

Question 290Question →

In $\triangle ABC$ , the lengths of sides $AB$ and $AC$ are both $13$ . A point $D$ lies on side $BC$ such that $AD$ is an integer. If the perimeter of $\triangle ABD$ is equal to the perimeter of $\triangle ACD$ , what is the sum of all possible integer values for the length of $BC$ ?

Show answer & explanation

Answer: 34

Answer

34

The perimeters of

\triangle ABD

and

\triangle ACD

are equal, and since

AB = AC = 13

, this forces

BD = CD

, making

D

the midpoint of

BC

. In the isosceles triangle

\triangle ABC

, the median

AD

is perpendicular to

BC

, making

\triangle ABD

a right triangle. By the Pythagorean theorem,

BD^2 + AD^2 = 169

. Since

AD

is an integer,

BD

must also be an integer (a half-integer would result in

AD^2

ending in

.25

, which cannot be a perfect square of an integer). The only positive integer solutions for

(BD, AD)

are

(5, 12)

and

(12, 5)

. This results in

BC = 2 \cdot BD

being either

10

or

24

. The sum of these possible values is

10 + 24 = 34

.

Step-by-Step Solution

1

Set the perimeters of

\triangle ABD

and

\triangle ACD

equal to each other.

BD = CD

Since

AB = AC = 13

, equating

AB + BD + AD = AC + CD + AD

simplifies directly to

BD = CD

.

2

Determine the relationship between

AD

and

BC

.

\triangle ABD

is a right triangle with hypotenuse

13

.

In an isosceles triangle, the median to the base is also the altitude, so

AD \perp BC

.

3

Apply the Pythagorean theorem to

\triangle ABD

.

BD^2 + AD^2 = 169

The sum of the squares of the legs in right triangle

\triangle ABD

must equal the square of the hypotenuse

AB = 13

.

4

Analyze the parity and integer constraints of

BD

and

AD

.

BD

must be a positive integer.

If

BD

were a half-integer,

BD^2

would end in

.25

, preventing

AD^2

from being an integer, which contradicts the given condition that

AD

is an integer.

5

Identify the Pythagorean triples with a hypotenuse of

13

.

(BD, AD) \in \{(5, 12), (12, 5)\}

The only positive integer solutions to

x^2 + y^2 = 13^2

are

(5, 12)

and

(12, 5)

.

6

Calculate the possible lengths of

BC

and sum them.

BC \in \{10, 24\}

, and their sum is

34

.

Since

D

is the midpoint of

BC

, the length of

BC

is

2 \cdot BD

, yielding

2 \cdot 5 = 10

and

2 \cdot 12 = 24

. Both satisfy the triangle inequality because

BC < AB + AC = 26

.

Key Concept

Properties of Isosceles Triangles and the Pythagorean Theorem

Estimated Time:3m 0s

Question 291Question →

A right triangle has two legs of equal length. If the hypotenuse of the triangle is $10\sqrt{2}$ centimeters, what is the length, in centimeters, of one of the legs?

Show answer & explanation

Answer: 10

Answer

The length of one of the legs is 10 centimeters.

An isosceles right triangle possesses acute angles of

45^\circ

and side ratios of

x : x : x\sqrt{2}

, where

x

represents the leg length. Given a hypotenuse of

10\sqrt{2}

centimeters, we equate

x\sqrt{2} = 10\sqrt{2}

. Dividing both sides of the equation by

\sqrt{2}

isolates the leg length, giving

x = 10

centimeters.

Step-by-Step Solution

1

Determine the triangle type from the given properties.

The triangle is a

45^\circ

-

45^\circ

-

90^\circ

special right triangle (isosceles right triangle).

A right triangle with two legs of equal length must have acute angles measuring

45^\circ

each, making it an isosceles right triangle.

2

Set up an equation utilizing the ratios of the side lengths.

Let

x

be the leg length. The hypotenuse length is represented by

x\sqrt{2} = 10\sqrt{2}

centimeters.

The hypotenuse of a

45^\circ

-

45^\circ

-

90^\circ

special right triangle is always

\sqrt{2}

times the length of one of its legs.

3

Solve the equation for the variable

x

.

x = 10

Dividing both sides of the equation by

\sqrt{2}

isolates the variable

x

representing the leg length.

Key Concept

Properties of

45^\circ

-

45^\circ

-

90^\circ

special right triangles.

Alternative Method

Alternatively, you can apply the Pythagorean Theorem:

a^2 + b^2 = c^2

. Since both legs are equal in length, we can set

a = b = x

. This yields the equation

x^2 + x^2 = (10\sqrt{2})^2

. Simplifying both sides gives

2x^2 = 100 \times 2 = 200

. Dividing by 2 yields

x^2 = 100

, and taking the square root of both sides gives

x = 10

centimeters.

Estimated Time:45s

Question 292Question →

If $8^{x - 2} = \left(\frac{1}{4}\right)^{x - 7}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is

4

.

To solve

8^{x - 2} = \left(\frac{1}{4}\right)^{x - 7}

, express both bases as powers of

2

. Since

8 = 2^3

and

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

, the equation can be written as

(2^3)^{x-2} = (2^{-2})^{x-7}

. Using the power of a power property

(a^m)^n = a^{mn}

, we simplify this to

2^{3x - 6} = 2^{-2x + 14}

. Because the bases are equal, their exponents must be equal:

3x - 6 = -2x + 14

. Adding

2x

and

6

to both sides yields

5x = 20

, which simplifies to

x = 4

.

Step-by-Step Solution

1

Express both bases as powers of 2

8 = 2^3

and

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

To solve exponential equations with different bases, rewrite them using a common base.

2

Substitute the powers back into the equation and simplify exponents

2^{3(x-2)} = 2^{-2(x-7)}

Apply the exponent power rule

(a^m)^n = a^{mn}

.

3

Equate the exponents since the bases are identical

3(x-2) = -2(x-7)

If

b^y = b^z

and

b > 0, b \neq 1

, then

y = z

.

4

Solve the linear equation for

x

3x - 6 = -2x + 14 \implies 5x = 20 \implies x = 4

Distribute, group like terms, and isolate the variable.

Key Concept

Solving exponential equations using a common base

Question 293Question →

A software company's weekly revenue, $R$ , in thousands of dollars, is modeled by the function $R(x) = -x^2 + 8x$ , where $x$ represents the price of the software in dollars. The company's weekly operating cost, $C$ , in thousands of dollars, is modeled by the function $C(x) = 2x + 5$ . The company breaks even when its weekly revenue equals its weekly operating cost. What is the sum of the possible software prices, in dollars, at which the company breaks even?

Show answer & explanation

Answer: 6

Answer

The sum of the possible software prices at which the company breaks even is 6.

To find the software prices at which the company breaks even, we set the revenue function

R(x)

equal to the cost function

C(x)

, which gives

-x^2 + 8x = 2x + 5

. Rearranging this equation into standard quadratic form gives

x^2 - 6x + 5 = 0

. Factoring this expression yields

(x - 5)(x - 1) = 0

, meaning the individual solutions are

x = 5

and

x = 1

. The sum of these possible break-even software prices is

5 + 1 = 6

.

Step-by-Step Solution

1

Set the revenue function equal to the cost function.

-x^2 + 8x = 2x + 5

The break-even points occur where the revenue and cost are equal.

2

Rearrange the terms to set the equation to zero.

x^2 - 6x + 5 = 0

Moving all terms to one side of the equation puts it in standard quadratic form, allowing us to solve for

x

.

3

Factor the quadratic equation.

(x - 5)(x - 1) = 0

Factoring is a standard method to find the roots of a quadratic equation of the form

ax^2 + bx + c = 0

.

4

Find the individual roots by setting each factor to zero.

x = 5

and

x = 1

If the product of two factors is zero, at least one of the factors must be zero.

5

Calculate the sum of the solutions.

5 + 1 = 6

The question specifically asks for the sum of the possible prices.

Key Concept

Solving systems of linear and non-linear equations by setting the equations equal to each other and solving the resulting quadratic equation.

Question 294Question →

A circle drawn in the standard $(x, y)$ coordinate plane has its center at the point $(3, -2)$ and passes through the point $(7, 1)$ . What is the radius of this circle?

Show answer & explanation

Answer: 5

Answer

The radius of the circle is 5.

The radius of a circle is defined as the distance from its center to any point on the boundary. Using the distance formula between the center

(3, -2)

and the point

(7, 1)

yields

r = \sqrt{(7 - 3)^2 + (1 - (-2))^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Identify the formula to find the distance between two points, which represents the radius of the circle.

The distance formula is

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

.

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

2

Substitute the coordinates of the center

(3, -2)

and the point

(7, 1)

into the formula.

r = \sqrt{(7 - 3)^2 + (1 - (-2))^2}

We substitute

x_1 = 3, y_1 = -2

and

x_2 = 7, y_2 = 1

into the distance formula.

3

Simplify the expression to determine the radius.

r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5

Calculate the squared differences, add them, and take the square root to find the final value.

Key Concept

Calculating the radius of a circle given its center and a point on the circle using the distance formula.

Question 295Question →

In $\triangle ABC$ , the measure of $\angle B$ is $80^\circ$ and the measure of $\angle C$ is $40^\circ$ . A point $D$ lies on side $BC$ such that $AD$ bisects $\angle BAC$ , and a point $E$ lies on side $AC$ such that $AD = AE$ . What is the measure, in degrees, of $\angle CDE$ ?

Show answer & explanation

Answer: 35

Answer

35

The correct answer is

35

. By first finding that

\angle BAC = 60^\circ

, we use the angle bisector

AD

to find

\angle CAD = 30^\circ

. In

\triangle ADC

, we find the interior angle

\angle ADC = 110^\circ

. In the isosceles triangle

\triangle ADE

with

AD=AE

, the base angles are

\angle ADE = \angle AED = 75^\circ

. Finally, subtracting

\angle ADE

from

\angle ADC

gives

\angle CDE = 35^\circ

.

Step-by-Step Solution

1

Find the measure of the third angle of the main triangle,

\angle BAC

.

\angle BAC = 60^\circ

The sum of the interior angles of any triangle is

180^\circ

. Therefore,

\angle BAC = 180^\circ - \angle B - \angle C = 180^\circ - 80^\circ - 40^\circ = 60^\circ

.

2

Determine the measure of the bisected angle

\angle CAD

.

\angle CAD = 30^\circ

Since

AD

bisects

\angle BAC

, it divides the angle into two equal parts:

\angle BAD = \angle CAD = \frac{60^\circ}{2} = 30^\circ

.

3

Calculate the interior angle

\angle ADC

in

\triangle ADC

.

\angle ADC = 110^\circ

In

\triangle ADC

, the sum of angles is

180^\circ

. Therefore,

\angle ADC = 180^\circ - \angle CAD - \angle C = 180^\circ - 30^\circ - 40^\circ = 110^\circ

.

4

Find the base angles of the isosceles triangle

ADE

.

\angle ADE = 75^\circ

Since

AD = AE

,

\triangle ADE

is an isosceles triangle with vertex angle

\angle DAE = 30^\circ

. The two base angles,

\angle ADE

and

\angle AED

, are equal. Thus,

\angle ADE = \frac{180^\circ - 30^\circ}{2} = 75^\circ

.

5

Determine the final angle

\angle CDE

by subtraction.

\angle CDE = 35^\circ

Since point

E

lies on side

AC

, ray

DE

lies between rays

DA

and

DC

. Therefore,

\angle ADC = \angle ADE + \angle CDE

. Rearranging gives

\angle CDE = \angle ADC - \angle ADE = 110^\circ - 75^\circ = 35^\circ

.

Key Concept

Applying triangle angle sum theorem, angle bisector properties, and isosceles triangle base angle properties to perform multi-step angle tracing.

Alternative Method

Use the exterior angle theorem on

\triangle ADC

at vertex

D

:

\angle ADB = \angle CAD + \angle C = 30^\circ + 40^\circ = 70^\circ

. Then, since

E

is on

AC

,

A

,

E

, and

C

are collinear. In

\triangle ADE

, the exterior angle at

E

is

\angle DEC = \angle DAE + \angle ADE = 30^\circ + 75^\circ = 105^\circ

. In

\triangle DEC

, the sum of angles is

180^\circ

, so

\angle CDE = 180^\circ - 105^\circ - 40^\circ = 35^\circ

.

Estimated Time:2m 30s

Question 296Question →

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the points $(a, a^2)$ and $(b, b^2)$ , where $a$ and $b$ are distinct real numbers. The slope of line $L$ is $8$ . If the midpoint of the line segment connecting these two points lies on the line $y = 5x - 1$ , what is the $y$ -coordinate of this midpoint?

Show answer & explanation

Answer: 19

Answer

The

y

-coordinate of the midpoint is

19

.

Applying the slope formula to the points

(a, a^2)

and

(b, b^2)

yields \frac{b^2-a^2}{b-a} = a+b = 8. The

x

-coordinate of the midpoint is \frac{a+b}{2} = 4. Substituting this value into the equation

y = 5x - 1

gives

y = 5(4) - 1 = 19

.

Step-by-Step Solution

1

Express the slope of line

L

in terms of

a

and

b

and set it equal to the given slope.

a + b = 8

The slope formula is

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

. For the points

(a, a^2)

and

(b, b^2)

, the slope is \frac{b^2 - a^2}{b - a}. Factoring the numerator gives \frac{(b-a)(b+a)}{b-a} = a + b. Since the slope is given as

8

, we establish

a + b = 8

.

2

Find the

x

-coordinate of the midpoint of the segment connecting the two points.

x_M = 4

The midpoint formula for the

x

-coordinate is

x_M = \frac{x_1 + x_2}{2}

. For our points,

x_M = \frac{a+b}{2}

. Substituting the value

a+b = 8

gives

x_M = \frac{8}{2} = 4

.

3

Determine the

y

-coordinate of the midpoint using the line equation.

y_M = 19

The midpoint lies on the line

y = 5x - 1

. Substituting

x_M = 4

into this equation gives

y_M = 5(4) - 1 = 19

.

Key Concept

Slope of a Line

Alternative Method

Let

a

and

b

be the real roots of the quadratic equation

t^2 - 8t + 13 = 0

. By Vieta's formulas,

a+b = 8

and

ab = 13

. The

x

-coordinate of the midpoint is \frac{a+b}{2} = 4, and the

y

-coordinate is \frac{a^2+b^2}{2} = \frac{(a+b)^2 - 2ab}{2} = \frac{64 - 26}{2} = 19. Since the point

(4, 19)

satisfies

y = 5x - 1

, this confirms the existence of valid real coordinates

(a, a^2)

and

(b, b^2)

that produce the midpoint on the line.

Estimated Time:2m 0s

Question 297Question →

A municipal water reservoir is being drained at a constant rate. After $4$ hours of draining, the reservoir contains $18,000$ gallons of water. After $7$ hours of draining, it contains $13,500$ gallons of water. How many hours after the draining process begins will the reservoir be completely empty?

Show answer & explanation

Answer: 16

Answer

The reservoir will be completely empty

16

hours after draining begins.

The correct answer is

16

hours. The constant draining rate is calculated as the change in volume divided by the change in time:

\frac{13,500 - 18,000}{7 - 4} = -1,500

gallons per hour. Since there are

18,000

gallons left at the

4

-hour mark, it will take an additional

\frac{18,000}{1,500} = 12

hours to completely empty the reservoir. The total time from the start is

4 + 12 = 16

hours.

Step-by-Step Solution

1

Calculate the draining rate (the slope of the linear function).

The rate is

-1,500

gallons per hour.

The rate of change is the change in volume divided by the change in time:

\frac{13,500 - 18,000}{7 - 4} = \frac{-4,500}{3} = -1,500

gallons per hour.

2

Determine the remaining time needed to empty the reservoir after the

7

-hour mark.

It will take an additional

9

hours.

At

7

hours, the reservoir contains

13,500

gallons. Draining at a rate of

1,500

gallons per hour, the remaining time is

\frac{13,500}{1,500} = 9

hours.

3

Calculate the total elapsed time since the draining process began.

16

hours

Adding the initial

7

hours of draining to the additional

9

hours needed gives

7 + 9 = 16

hours.

Key Concept

Linear word problems and finding intercepts

Alternative Method

We can model the volume of water

W

as a linear function of time

t

using the slope-intercept form

W(t) = mt + b

. Substituting the rate

m = -1,500

and the point

(4, 18,000)

gives

18,000 = -1,500(4) + b

, which yields the

y

-intercept (initial volume)

b = 24,000

gallons. The linear model is

W(t) = -1,500t + 24,000

. Setting

W(t) = 0

to find when the reservoir is empty gives

0 = -1,500t + 24,000

, which simplifies to

t = \frac{24,000}{1,500} = 16

hours.

Estimated Time:1m 30s

Question 298Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $(x - 9)^2 + (y + 4)^2 = 81$ . What is the radius of this circle?

Show answer & explanation

Answer: 9

Answer

The radius of the circle is 9.

The standard equation of a circle in the coordinate plane is

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

, where

r

is the radius. For the equation

(x - 9)^2 + (y + 4)^2 = 81

, the right-hand side represents

r^2

, so

r^2 = 81

. Taking the positive square root of both sides gives

r = \sqrt{81} = 9

.

Step-by-Step Solution

1

Identify the standard form of the circle equation.

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 allows us to relate the terms of the given equation to the components of the circle.

2

Match the given equation to the standard form.

By comparing

(x - 9)^2 + (y + 4)^2 = 81

to

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

, we find that

r^2 = 81

.

The constant on the right side of the standard equation represents the square of the radius.

3

Solve for the radius

r

.

r = \sqrt{81} = 9

.

Taking the square root of

r^2

gives the actual radius of the circle.

Key Concept

Identifying the radius from the standard equation of a circle

Question 299Question →

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 300Question →

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

Answer

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Key Concept

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