All practice questions

Question 2321Question →

A circle has a center located at the point $(-2, 5)$ and a radius of $6$ units. What is the equation of this circle in the $(x, y)$ coordinate plane?

(x + 2)^2 + (y - 5)^2 = 36

(x - 2)^2 + (y + 5)^2 = 36

(x + 2)^2 + (y - 5)^2 = 6

(x - 2)^2 + (y - 5)^2 = 36

(x - 2)^2 + (y + 5)^2 = 6

Show answer & explanation

Answer: $(x + 2)^2 + (y - 5)^2 = 36$

Answer

The equation of the circle is

(x + 2)^2 + (y - 5)^2 = 36

.

The correct equation is found by substituting the center

(-2, 5)

and radius

6

into the standard equation

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

. This yields

(x - (-2))^2 + (y - 5)^2 = 6^2

, which simplifies to

(x + 2)^2 + (y - 5)^2 = 36

.

Step-by-Step Solution

1

Identify the standard form equation of a circle.

The standard equation is

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

, where

(h, k)

represents the center coordinates and

r

represents the radius.

This general equation is the mathematical template for representing any circle in a coordinate plane.

2

Substitute the given values for the center

(-2, 5)

and the radius

6

into the equation.

(x - (-2))^2 + (y - 5)^2 = 6^2

Plugging in

h = -2

,

k = 5

, and

r = 6

customizes the general formula to this specific circle.

3

Simplify the signs of the terms and calculate the squared radius.

(x + 2)^2 + (y - 5)^2 = 36

Subtracting

-2

simplifies to adding

2

, and squaring the radius of

6

yields

36

.

Key Concept

Standard equation of a circle given its center and radius

Question 2322Question →

A laboratory technician mixes two solutions. The volume of the first solution is represented by $\frac{1}{2}(x - 5)$ liters, and the volume of the second solution is represented by $\frac{1}{3}(2x + 1)$ liters, where $x$ is a positive real number. If the sum of the volumes of these two solutions is $6$ liters, what is the value of $3x - 4$ ?

6

9

17

30

53

Show answer & explanation

Answer: $17$

Answer

The value of the expression is

17

.

The sum of the volumes of the two solutions is

6

liters, which translates to the linear equation

\frac{1}{2}(x - 5) + \frac{1}{3}(2x + 1) = 6

. Multiplying the entire equation by the least common multiple of the denominators,

6

, yields

3(x - 5) + 2(2x + 1) = 36

. Distributing the constants leads to

3x - 15 + 4x + 2 = 36

. Combining like terms gives

7x - 13 = 36

. Adding

13

to both sides results in

7x = 49

, which gives

x = 7

. Substituting

x = 7

into the expression

3x - 4

yields

3(7) - 4 = 17

, which is the correct value.

Step-by-Step Solution

1

Set up the linear equation based on the word problem context.

\frac{1}{2}(x - 5) + \frac{1}{3}(2x + 1) = 6

The sum of the volumes of the two solutions is given as

6

liters.

2

Clear the fractions by multiplying both sides of the equation by the least common multiple of the denominators.

3(x - 5) + 2(2x + 1) = 36

Multiplying by

6

eliminates the fractions and simplifies the equation for solving.

3

Distribute the coefficients to eliminate the parentheses.

3x - 15 + 4x + 2 = 36

Applying the distributive property allows like terms to be grouped.

4

Combine like terms and isolate the variable

x

.

7x - 13 = 36 \implies 7x = 49 \implies x = 7

Adding

13

to both sides and dividing by

7

solves for the value of the variable

x

.

5

Evaluate the expression requested in the question using the solved value of

x

.

3(7) - 4 = 21 - 4 = 17

Substituting

x = 7

into

3x - 4

yields the final requested value.

Key Concept

Solving linear equations with fractional coefficients by clearing denominators and evaluating variable expressions.

Alternative Method

Instead of clearing the fractions immediately, we can distribute the fractions first:

\frac{1}{2}x - 2.5 + \frac{2}{3}x + \frac{1}{3} = 6

. Converting to fractions with a common denominator of 6, we get

\frac{3}{6}x - \frac{15}{6} + \frac{4}{6}x + \frac{2}{6} = 6

, which simplifies to

\frac{7}{6}x - \frac{13}{6} = 6

. Adding

\frac{13}{6}

to both sides yields

\frac{7}{6}x = \frac{49}{6}

, so

7x = 49

, and

x = 7

. Substituting

x = 7

into the expression

3x - 4

yields

17

.

Estimated Time:1m 30s

Question 2323Question →

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 2324Question →

In the standard $(x, y)$ coordinate plane, line $l_1$ passes through the points $(1, 2)$ and $(5, r)$ , and line $l_2$ passes through the points $(-1, r)$ and $(3, 8)$ . If the slope of line $l_2$ is $-2$ times the slope of line $l_1$ , what is the value of $r$ ?

6

4

-4

-\frac{4}{3}

14

Show answer & explanation

Answer: $-4$

Answer

The value of

r

is

-4

.

The correct value of

r

is

-4

. The slope of line

l_1

is

m_1 = \frac{r - 2}{4}

and the slope of line

l_2

is

m_2 = \frac{8 - r}{4}

. Given that

m_2 = -2m_1

, substituting these expressions yields

\frac{8 - r}{4} = -2\left(\frac{r - 2}{4}\right)

. Multiplying both sides by

4

gives

8 - r = -2r + 4

. Adding

2r

to both sides and subtracting

8

from both sides results in

r = -4

.

Step-by-Step Solution

1

Find the slope

m_1

of line

l_1

using the coordinates

(1, 2)

and

(5, r)

.

m_1 = \frac{r - 2}{5 - 1} = \frac{r - 2}{4}

The slope formula is

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

.

2

Find the slope

m_2

of line

l_2

using the coordinates

(-1, r)

and

(3, 8)

.

m_2 = \frac{8 - r}{3 - (-1)} = \frac{8 - r}{4}

Applying the slope formula to the points on the second line.

3

Set up the equation using the given relationship

m_2 = -2m_1

.

\frac{8 - r}{4} = -2\left(\frac{r - 2}{4}\right)

We are given that the slope of line

l_2

is

-2

times the slope of line

l_1

.

4

Solve the equation for

r

.

8 - r = -2(r - 2) \Rightarrow 8 - r = -2r + 4 \Rightarrow r = -4

Multiply both sides by

4

and simplify to solve for

r

.

Key Concept

Slope of a Line

Alternative Method

Instead of solving the equation algebraically, you can test the given choices for

r

by calculating the slopes

m_1

and

m_2

for each option and checking if

m_2 = -2m_1

.

Estimated Time:2m 0s

Question 2325Question →

In the standard $(x, y)$ coordinate plane, a triangle undergoes a sequence of transformations. First, it is reflected across the line $y = -x$ . Then, the resulting figure is translated $3$ units to the right and $2$ units down. If the vertex $A''$ of the final image is located at $(-1, 5)$ , what were the coordinates of the original vertex $A$ ?

(-7, 4)

(-3, -2)

(7, -4)

(4, -7)

(-4, 7)

Show answer & explanation

Answer: $(-7, 4)$

Answer

The original coordinates of vertex

A

are

(-7, 4)

.

The coordinates of the original vertex are found by reversing the sequence of transformations. Starting from the final image at

(-1, 5)

, we first reverse the translation by moving

3

units to the left and

2

units up, which gives the intermediate point

(-4, 7)

. Then, we apply the reflection across the line

y = -x

to this intermediate point. The reflection rule swaps the coordinates and negates them, mapping

(-4, 7)

to

(-7, 4)

.

Step-by-Step Solution

1

Identify the forward transformation rules for a point

(x, y)

.

The reflection across

y = -x

maps

(x, y) \rightarrow (-y, -x)

. The translation

3

units right and

2

units down maps

(x', y') \rightarrow (x' + 3, y' - 2)

.

Understanding the forward transformations allows us to set up the equations or work backward systematically.

2

Work backward from the final image

A''(-1, 5)

by reversing the translation.

Reversing a translation of

3

units right and

2

units down means translating

3

units left and

2

units up. Applying this to

A''(-1, 5)

yields

(-1 - 3, 5 + 2) = (-4, 7)

.

This gives the intermediate coordinates of the point

A'

before the translation occurred.

3

Reverse the reflection across the line

y = -x

.

A reflection across

y = -x

is its own inverse. Applying the mapping

(x', y') \rightarrow (-y', -x')

to the intermediate point

(-4, 7)

yields

(-7, 4)

.

This step recovers the original pre-image coordinates of vertex

A

.

Key Concept

Reversing composite transformations in the coordinate plane

Estimated Time:1m 30s

Question 2326Question →

If $x$ and $y$ are positive real numbers such that $\log(x + y) = \log x + \log y$ , which of the following equations expresses $y$ in terms of $x$ ?

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

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

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

(where

x > 1

)

y = x

Any positive real number

y

, because the equation is a general identity for all positive

x

and

y

Show answer & explanation

Answer: $y = \frac{x}{x-1}$ (where $x > 1$ )

Answer

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

(where

x > 1

)

By applying the product property of logarithms, the right side of the given equation

\log x + \log y

simplifies to

\log(xy)

. This transforms the equation to

\log(x+y) = \log(xy)

. Equating the arguments gives the algebraic equation

x+y=xy

. Rearranging this equation to group all

y

terms on one side yields

x = xy - y

. Factoring out

y

gives

x = y(x-1)

. Finally, dividing both sides by

x-1

results in the equation stating that

y

is equal to

x

divided by the quantity

x

minus one.

Step-by-Step Solution

1

Apply the product property of logarithms to the right side of the equation:

\log x + \log y = \log(xy)

.

The equation becomes

\log(x + y) = \log(xy)

.

This allows both sides to be expressed as a single logarithm with the same base.

2

Equate the arguments of the logarithms, since

\log(u) = \log(v)

implies

u = v

for positive real numbers.

x + y = xy

To eliminate the logarithms and work with a purely algebraic equation.

3

Isolate the terms containing

y

on one side of the equation.

x = xy - y

Grouping the

y

terms on the same side is a necessary step to factor out the variable we want to solve for.

4

Factor out

y

from the right side and divide both sides by the remaining coefficient

x - 1

.

x = y(x - 1) \implies y = \frac{x}{x-1}

(for

x > 1

to keep

y

positive and avoid division by zero).

This completely isolates

y

in terms of

x

.

Key Concept

Properties of Logarithms and Algebraic Isolation

Question 2327Question →

In a Cartesian coordinate system, the point $A(2, -4)$ is translated $4$ units to the left and $3$ units up to form point $B$ . If point $B$ is then reflected across the line $y = -x$ to form point $C$ , what are the coordinates of point $C$ ?

(1, 2)

(-1, -2)

(7, -6)

(7, 2)

(-2, 1)

Show answer & explanation

Answer: $(1, 2)$

Answer

(1, 2)

First, the point

(2, -4)

is translated

4

units left and

3

units up by subtracting

4

from the

x

-coordinate and adding

3

to the

y

-coordinate, which yields

(-2, -1)

. Next, this point is reflected across the line

y = -x

by swapping the coordinates and negating both, which results in the point

(1, 2)

.

Step-by-Step Solution

1

Apply the translation to point

A(2, -4)

.

Point

B

has coordinates

(-2, -1)

.

Translating a point

4

units to the left subtracts

4

from the

x

-coordinate, and translating

3

units up adds

3

to the

y

-coordinate:

(2 - 4, -4 + 3) = (-2, -1)

.

2

Apply the reflection across the line

y = -x

to point

B(-2, -1)

.

Point

C

has coordinates

(1, 2)

.

Reflecting a point

(x, y)

across the line

y = -x

swaps the coordinates and negates both, mapping

(x, y) \rightarrow (-y, -x)

. Thus,

(-2, -1) \rightarrow (-(-1), -(-2)) = (1, 2)

.

Key Concept

Composite transformations involving translations and reflections in the coordinate plane.

Question 2328Question →

A square tabletop has a diagonal length of $8$ feet. What is the area of the tabletop, in square feet?

16

16\sqrt{2}

32

32\sqrt{2}

64

Show answer & explanation

Answer: 32

Answer

32

The correct answer is 32. The diagonal of a square divides it into two congruent 45-45-90 right triangles, where the diagonal is the hypotenuse. The ratio of the diagonal to the side length in a 45-45-90 triangle is

\sqrt{2}

to

1

. Therefore, a square with a diagonal of

8

feet has a side length of

s = \frac{8}{\sqrt{2}}

feet. The area of the square is

s^2 = \left(\frac{8}{\sqrt{2}}\right)^2 = \frac{64}{2} = 32

square feet.

Step-by-Step Solution

1

Relate the diagonal of a square to its side length using special right triangles.

The diagonal of a square splits the square into two 45-45-90 right triangles. The hypotenuse of these triangles is the diagonal,

8

feet, and the legs are the sides of the square,

s

. The relationship is

s\sqrt{2} = 8

.

In a 45-45-90 triangle, the hypotenuse is

\sqrt{2}

times the length of a leg.

2

Solve for the side length

s

of the square tabletop.

s = \frac{8}{\sqrt{2}}

feet.

Divide both sides of the equation by

\sqrt{2}

to isolate the side length

s

.

3

Calculate the area of the square tabletop.

Area

= s^2 = \left(\frac{8}{\sqrt{2}}\right)^2 = \frac{64}{2} = 32

square feet.

The area of a square is calculated by squaring its side length.

Key Concept

Using the properties of 45-45-90 special right triangles to find side lengths and area from the diagonal of a square.

Question 2329Question →

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 2330Question →

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 2331Question →

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 2332Question →

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 2333Question →

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 2334Question →

An infinite geometric series has a first term of $12$ and a sum of $8$ . What is the common ratio of this series?

-\frac{1}{2}

\frac{1}{2}

\frac{1}{3}

\frac{2}{3}

\frac{5}{2}

Show answer & explanation

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

Answer

The correct common ratio is

-\frac{1}{2}

.

The correct answer is

-\frac{1}{2}

. The formula for the sum of an infinite geometric series is

S = \frac{a_1}{1 - r}

. Substituting the first term

a_1 = 12

and the sum

S = 8

gives

8 = \frac{12}{1 - r}

. Multiplying both sides by

1 - r

yields

8(1 - r) = 12

, which simplifies to

8 - 8r = 12

. Subtracting

8

from both sides gives

-8r = 4

. Dividing by

-8

gives

r = -\frac{1}{2}

. Since

|-\frac{1}{2}| < 1

, the series converges.

Step-by-Step Solution

1

Write the formula for the sum of an infinite geometric series.

S = \frac{a_1}{1 - r}

This formula relates the sum (

S

), the first term (

a_1

), and the common ratio (

r

) of an infinite geometric series where

|r| < 1

.

2

Substitute the given values into the formula.

8 = \frac{12}{1 - r}

Replacing

S

with

8

and

a_1

with

12

leaves the common ratio

r

as the only unknown variable.

3

Solve for the common ratio

r

.

8(1 - r) = 12 \Rightarrow 8 - 8r = 12 \Rightarrow -8r = 4 \Rightarrow r = -\frac{4}{8} = -\frac{1}{2}

Multiplying by the denominator and isolating

r

yields the value of the common ratio.

Key Concept

Sum of an infinite geometric series

Alternative Method

Test the given choices by substituting each value of

r

back into the sum formula

S = \frac{12}{1 - r}

to see which one yields a sum of

8

. For example, testing

-\frac{1}{2}

gives

\frac{12}{1 - (-0.5)} = \frac{12}{1.5} = 8

, which matches the given sum.

Estimated Time:1m 0s

Question 2335Question →

A scientist is monitoring the temperature of a chemical solution. At time $t = \frac{1}{3}$ hours, the temperature is $2^\circ\text{C}$ . At time $t = \frac{1}{2}$ hours, the temperature is $5^\circ\text{C}$ . Assuming the temperature increases at a constant rate, what was the initial temperature of the solution, in degrees Celsius?

-4

\frac{2}{9}

3

8

30

Show answer & explanation

Answer: -4

Answer

-4

The linear relationship is modeled by

T(t) = mt + b

. Calculating the slope

m

gives

\frac{5 - 2}{1/2 - 1/3} = \frac{3}{1/6} = 18

. Substituting the point

(1/3, 2)

into the equation yields

2 = 18(1/3) + b

, which simplifies to

2 = 6 + b

, giving

b = -4

. Thus, the initial temperature at

t = 0

is

-4^\circ\text{C}

.

Step-by-Step Solution

1

Find the constant rate of change (slope,

m

) using the points

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

and

(\frac{1}{2}, 5)

.

m = \frac{5 - 2}{\frac{1}{2} - \frac{1}{3}} = \frac{3}{\frac{1}{6}} = 18

The rate of change represents the slope of the linear relationship between time and temperature.

2

Use the slope-intercept form

T(t) = mt + b

and the point

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

to solve for the vertical intercept (

b

), which represents the initial temperature.

2 = 18(\frac{1}{3}) + b \Rightarrow 2 = 6 + b \Rightarrow b = -4

The initial temperature occurs at time

t = 0

, which corresponds to the

y

-intercept (

b

) of the linear equation.

Key Concept

Determining a linear equation from two points to find the initial value (vertical intercept)

Question 2336Question →

What is the slope of the line represented by the equation $5x + 3y = -12$ in the standard $(x, y)$ coordinate plane?

\frac{5}{3}

-\frac{3}{5}

-\frac{5}{3}

\frac{3}{5}

-5

Show answer & explanation

Answer: $-\frac{5}{3}$

Answer

-\frac{5}{3}

The correct answer is found by converting the standard form equation

5x + 3y = -12

into slope-intercept form (

y = mx + b

). Subtracting

5x

from both sides gives

3y = -5x - 12

. Dividing both sides by

3

yields

y = -\frac{5}{3}x - 4

. Thus, the slope of the line is the coefficient of

x

, which is

-\frac{5}{3}

.

Step-by-Step Solution

1

Start with the given equation in standard form.

5x + 3y = -12

Identify the equation to be transformed.

2

Subtract

5x

from both sides of the equation to isolate the term containing

y

.

3y = -5x - 12

Move the

x

term to the right side of the equation.

3

Divide every term in the equation by

3

to solve for

y

in slope-intercept form (

y = mx + b

).

y = -\frac{5}{3}x - 4

Isolate

y

so that the coefficient of

x

represents the slope.

4

Identify the coefficient of

x

, which is the slope

m

.

m = -\frac{5}{3}

In the form

y = mx + b

, the slope is

m

.

Key Concept

To find the slope of a line from its linear equation, rewrite the equation in slope-intercept form,

y = mx + b

, where

m

represents the slope.

Question 2337Question →

In the standard $(x, y)$ coordinate plane, a triangle has vertex $A(3, -4)$ . If the triangle is reflected across the line $y = x$ and then translated $5$ units to the left and $2$ units up, what are the coordinates of the image of vertex $A$ ?

(1, 1)

(-2, -2)

(-2, 6)

(-9, 5)

(-1, -1)

Show answer & explanation

Answer: $(-9, 5)$

Answer

(-9, 5)

To find the coordinates of the image, we apply the transformations in the specified order. First, reflecting the point

(3, -4)

across the line

y = x

swaps the

x

- and

y

-coordinates, resulting in

(-4, 3)

. Second, translating the point

5

units to the left subtracts

5

from the

x

-coordinate, and translating

2

units up adds

2

to the

y

-coordinate:

(-4 - 5, 3 + 2) = (-9, 5)

.

Step-by-Step Solution

1

Reflect the point across the line

y = x

(-4, 3)

The coordinate rule for a reflection across the line

y = x

is

(x, y) \rightarrow (y, x)

. Applying this to the vertex

A(3, -4)

swaps the coordinates to yield

(-4, 3)

.

2

Translate the point

5

units to the left and

2

units up

(-9, 5)

Translating a point

5

units left subtracts

5

from the

x

-coordinate, and translating

2

units up adds

2

to the

y

-coordinate:

(-4 - 5, 3 + 2) = (-9, 5)

.

Key Concept

Applying composite transformations in the coordinate plane including reflections and translations.

Estimated Time:1m 0s

Question 2338Question →

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 2339Question →

A geometric sequence consists of positive terms. The first term of the sequence is $3$ , and the sum of the first $3$ terms is $39$ . What is the value of the $4$ th term of this sequence?

33

81

120

192

243

Show answer & explanation

Answer: 81

Answer

81

The sum of the first three terms of a geometric sequence is given by

S_3 = a_1(1 + r + r^2)

. Substituting the first term

a_1 = 3

and the sum

S_3 = 39

gives

3(1 + r + r^2) = 39

, which simplifies to

r^2 + r - 12 = 0

. Factoring this quadratic equation yields

(r - 3)(r + 4) = 0

. Since the sequence consists of positive terms, the common ratio must be positive, which gives

r = 3

. The fourth term of the sequence is then calculated using the formula

a_4 = a_1 r^3 = 3 \cdot 3^3 = 81

.

Step-by-Step Solution

1

Write the expression for the sum of the first 3 terms of a geometric sequence and set it equal to the given sum.

S_3 = a_1(1 + r + r^2) = 3(1 + r + r^2) = 39

This sets up the equation needed to solve for the common ratio of the sequence.

2

Divide both sides of the equation by 3 and solve the resulting quadratic equation for the common ratio

r

.

1 + r + r^2 = 13 \implies r^2 + r - 12 = 0 \implies (r - 3)(r + 4) = 0 \implies r = 3

(since terms must be positive)

This determines the common ratio of the geometric sequence.

3

Use the formula for the

n

-th term of a geometric sequence,

a_n = a_1 r^{n-1}

, to find the value of the 4th term.

a_4 = 3 \cdot 3^3 = 3 \cdot 27 = 81

This computes the final value requested by the question.

Key Concept

Geometric sequence formulas for the sum of a finite number of terms and the n-th term

Estimated Time:1m 30s

Question 2340Question →

In the standard $(x, y)$ coordinate plane, the perpendicular bisector of the line segment with endpoints $D(-5, 8)$ and $E(3, 4)$ is represented by the equation $y = mx + b$ . What is the value of $m + b$ ?

2

\frac{7}{2}

5

7

10

Show answer & explanation

Answer: $10$

Answer

The sum of the slope and the y-intercept of the perpendicular bisector is

10

.

The slope of the segment

DE

is calculated as

m_{DE} = \frac{4 - 8}{3 - (-5)} = -\frac{1}{2}

. The perpendicular bisector has a slope

m

that is the negative reciprocal of this, which is

2

. The line must pass through the midpoint of

DE

, which is located at

\left(\frac{-5 + 3}{2}, \frac{8 + 4}{2}\right) = (-1, 6)

. Substituting

m = 2

and the point

(-1, 6)

into the slope-intercept form

y = mx + b

yields

6 = 2(-1) + b

, which simplifies to

b = 8

. Summing the slope and the

y

-intercept gives

m + b = 2 + 8 = 10

.

Step-by-Step Solution

1

Find the midpoint of the line segment

DE

.

Midpoint

M = \left(\frac{-5 + 3}{2}, \frac{8 + 4}{2}\right) = (-1, 6)

By definition, the perpendicular bisector must pass through the midpoint of the segment it bisects.

2

Calculate the slope of the line segment

DE

.

m_{DE} = \frac{4 - 8}{3 - (-5)} = \frac{-4}{8} = -\frac{1}{2}

The slope of the segment is needed to find the slope of any line perpendicular to it.

3

Determine the slope

m

of the perpendicular bisector.

m = -\frac{1}{m_{DE}} = -\frac{1}{-1/2} = 2

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

4

Find the

y

-intercept

b

of the perpendicular bisector.

Using the slope-intercept form

y = mx + b

with the slope

m = 2

and the midpoint

M(-1, 6)

:

6 = 2(-1) + b \implies 6 = -2 + b \implies b = 8

Substituting a known point on the line allows us to solve for the vertical intercept parameter.

5

Calculate the sum

m + b

.

m + b = 2 + 8 = 10

This is the value requested by the question.

Key Concept

The perpendicular bisector of a line segment passes through its midpoint at a right angle, meaning its slope is the negative reciprocal of the segment's slope.

Estimated Time:2m 0s

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