All practice questions

Question 241Question →

In the quadratic equation $x^2 - bx + 64 = 0$ , $b$ is a constant. The two real solutions to the equation are $r$ and $s$ , where $r = s^2$ . What is the value of $b$ ?

Show answer & explanation

Answer: 20

Answer

20

By applying Vieta's formulas to the quadratic equation

x^2 - bx + 64 = 0

, we find that the product of the solutions is

r \cdot s = 64

. Substituting the given relation

r = s^2

gives

s^3 = 64

, which yields the real solution

s = 4

. Substituting this back into the relation gives the other solution

r = 16

. Finally, the sum of the solutions is

r + s = b

, so

b = 16 + 4 = 20

.

Step-by-Step Solution

1

Apply Vieta's formula for the product of the roots

r \cdot s = 64

For a quadratic equation in the form

x^2 - bx + c = 0

, the product of the roots is equal to the constant term

c

.

2

Substitute the given root relationship into the product equation

s^3 = 64

, which solves to

s = 4

We are given that one root is the square of the other (

r = s^2

), so substituting

s^2

for

r

allows us to solve for

s

.

3

Calculate the value of the second root

r

r = 16

Using the relation

r = s^2

with

s = 4

, we find

r = 4^2 = 16

.

4

Apply Vieta's formula for the sum of the roots to find

b

b = 20

For the equation

x^2 - bx + 64 = 0

, the sum of the roots is

r + s = b

. Substituting

r = 16

and

s = 4

gives

16 + 4 = 20

.

Key Concept

Vieta's formulas relating the coefficients of a quadratic equation to its roots

Question 242Question →

If $4^x = 5$ , what is the value of $2^{4x - 1}$ ?

Show answer & explanation

Answer: 12.5

Answer

12.5

By writing

4^x

as

(2^2)^x = 2^{2x}

, we find that

2^{2x} = 5

. The expression

2^{4x-1}

can be rewritten using exponent properties as

\frac{2^{4x}}{2^1} = \frac{(2^{2x})^2}{2}

. Substituting

2^{2x} = 5

into this expression yields

\frac{5^2}{2} = \frac{25}{2} = 12.5

. Thus, the correct numerical response is 12.5.

Step-by-Step Solution

1

Express the given equation in terms of base 2.

2^{2x} = 5

Since

4 = 2^2

, we can rewrite

4^x

as

(2^2)^x

. Applying the power of a power rule,

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

, gives

(2^2)^x = 2^{2x}

.

2

Rewrite the expression to be evaluated using exponent properties.

2^{4x - 1} = \frac{(2^{2x})^2}{2}

Using the division property of exponents,

a^{m-n} = \frac{a^m}{a^n}

, we can write

2^{4x - 1}

as

\frac{2^{4x}}{2}

. Then, using the power of a power rule in reverse,

2^{4x} = (2^{2x})^2

.

3

Substitute the known value of

2^{2x}

and simplify the numerical expression.

12.5

Substitute

2^{2x} = 5

into the expression

\frac{(2^{2x})^2}{2}

to obtain

\frac{5^2}{2} = \frac{25}{2}

, which simplifies to

12.5

.

Key Concept

Manipulating exponential equations by expressing bases in terms of common prime factors and applying properties of exponents.

Estimated Time:1m 30s

Question 243Question →

In the quadratic equation $x^2 - ax + b = 0$ , the roots $r$ and $s$ are positive integers. The quadratic equation $x^2 - (a+12)x + (b+12) = 0$ has roots $r^2$ and $s^2$ , where $a$ and $b$ are constants. What is the value of $a$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a

is 5.

Applying Vieta's formulas to both equations and using the algebraic identity

r^2 + s^2 = (r+s)^2 - 2rs

yields the system of equations

a^2 - 2b = a + 12

and

b^2 = b + 12

. Solving the second equation gives

b = 4

(discarding the negative root since

r, s > 0

). Substituting

b = 4

into the first equation yields

a^2 - a - 20 = 0

, which solves to

a = 5

(discarding

a = -4

since

r, s > 0

).

Step-by-Step Solution

1

Apply Vieta's formulas to the first quadratic equation.

r+s = a

and

rs = b

To express the sum and product of the roots of the first equation in terms of the coefficients

a

and

b

.

2

Apply Vieta's formulas to the second quadratic equation.

r^2+s^2 = a+12

and

r^2s^2 = b+12

To express the sum and product of the squared roots in terms of the coefficients of the second equation.

3

Express

r^2+s^2

in terms of

a

and

b

using the algebraic expansion of a binomial square.

a^2 - 2b = a+12

By rewriting

r^2+s^2

as

(r+s)^2 - 2rs

, we can substitute

a

and

b

to form a system of equations.

4

Solve for the constant

b

using the product relation.

b = 4

The relation

(rs)^2 = b^2 = b+12

simplifies to the quadratic equation

b^2 - b - 12 = 0

. Factoring yields

b = 4

or

b = -3

. Since

r

and

s

are positive integers, their product

b

must be positive, meaning

b = 4

.

5

Substitute

b = 4

into the sum relation to solve for

a

.

a = 5

Substituting

b = 4

yields

a^2 - 2(4) = a+12

, which simplifies to

a^2 - a - 20 = 0

. Factoring yields

a = 5

or

a = -4

. Since

r

and

s

are positive integers, their sum

a

must be positive, meaning

a = 5

.

Key Concept

Vieta's Formulas and Symmetric Polynomial Relationships

Estimated Time:3m 0s

Question 244Question →

A coffee shop sells two sizes of bagged coffee: a standard bag and a deluxe bag. On Monday, the shop sold $8$ standard bags and $5$ deluxe bags for a total of $\$108$ . On Tuesday, the shop sold $6$ standard bags and $10$ deluxe bags for a total of $\$156$ . What is the price, in dollars, of one deluxe bag of coffee?

Show answer & explanation

Answer: 12

Answer

The price of one deluxe bag of coffee is

12

dollars.

By setting up a system of equations, we represent the total cost on Monday as

8s + 5d = 108

and on Tuesday as

6s + 10d = 156

. Multiplying the first equation by

2

gives

16s + 10d = 216

. Subtracting the second equation yields

10s = 60

, so

s = 6

. Substituting

s = 6

into the first equation yields

8(6) + 5d = 108

, which simplifies to

48 + 5d = 108

, then

5d = 60

, and finally

d = 12

.

Step-by-Step Solution

1

Define the variables and write the system of equations.

Let

s

be the price of a standard bag and

d

be the price of a deluxe bag. The system is:

8s + 5d = 108

6s + 10d = 156

To translate the word problem into mathematical equations.

2

Multiply the first equation by

2

to align the coefficients of

d

.

16s + 10d = 216

To prepare for elimination of the variable

d

.

3

Subtract the second equation from the new equation to solve for

s

.

10s = 60

, which simplifies to

s = 6

.

To eliminate

d

and find the price of a standard bag.

4

Substitute the value of

s

back into one of the original equations to solve for

d

.

8(6) + 5d = 108 \implies 48 + 5d = 108 \implies 5d = 60 \implies d = 12

.

To find the price of a deluxe bag.

Key Concept

Solving systems of linear equations using the elimination method.

Estimated Time:1m 30s

Question 245Question →

An exponential function $g$ is defined by $g(x) = p \cdot q^x$ , where $p$ and $q$ are positive constants. If $g\left(\frac{1}{2}\right) = 6$ and $g\left(\frac{3}{2}\right) = 54$ , what is the value of $g(2)$ ?

Show answer & explanation

Answer: 162

Answer

162

The correct answer is 162. Substituting the coordinates into the function gives the system

p \cdot q^{1/2} = 6

and

p \cdot q^{3/2} = 54

. Dividing the second equation by the first yields

q = 9

. Substituting

q = 9

into the first equation yields

p \cdot 3 = 6

, so

p = 2

. The complete function is

g(x) = 2 \cdot 9^x

, and evaluating

g(2)

gives

2 \cdot 81 = 162

.

Step-by-Step Solution

1

Substitute the given values into the function definition to set up a system of equations.

p \cdot q^{1/2} = 6

and

p \cdot q^{3/2} = 54

To represent the given coordinate relationships using the exponential model parameters.

2

Divide the equation for

g\left(\frac{3}{2}\right)

by the equation for

g\left(\frac{1}{2}\right)

to eliminate

p

and solve for

q

.

q = 9

Using the exponent quotient rule,

\frac{q^{3/2}}{q^{1/2}} = q^{3/2 - 1/2} = q^1 = q

. Dividing the constants gives

\frac{54}{6} = 9

.

3

Substitute

q = 9

back into the first equation and solve for

p

.

p = 2

Evaluating the fractional exponent gives

9^{1/2} = \sqrt{9} = 3

. Substituting this in yields

3p = 6

, which simplifies to

p = 2

.

4

Evaluate the completed exponential function

g(x) = 2 \cdot 9^x

at

x = 2

.

162

Calculating

g(2) = 2 \cdot 9^2 = 2 \cdot 81 = 162

solves for the target value.

Key Concept

Solving a system of exponential equations using fractional exponent rules and base evaluation.

Question 246Question →

A geophysicist models the temperature, $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of a rock layer during a deep-crust drilling project using a linear function of the depth, $d$ , in kilometers ( $\text{km}$ ), below the surface. According to the model, for every increase in depth of $0.8\text{ km}$ , the temperature of the rock increases by $22^\circ\text{C}$ . At a depth of $2.4\text{ km}$ , the temperature of the rock is $81^\circ\text{C}$ . According to the model, at what depth, in kilometers, will the temperature of the rock be $114^\circ\text{C}$ ?

Show answer & explanation

Answer: 3.6

Answer

The temperature of the rock will reach

114^\circ\text{C}

at a depth of

3.6

kilometers.

The correct answer is

3.6

. The rate of change of temperature with depth is

\frac{22^\circ\text{C}}{0.8\text{ km}} = 27.5^\circ\text{C/km}

. The linear relationship between temperature

T

and depth

d

can be modeled by

T = 27.5d + T_0

, where

T_0

is the temperature at the surface. Substituting the known values

d = 2.4

and

T = 81

into the model gives

81 = 27.5(2.4) + T_0

, which simplifies to

81 = 66 + T_0

. Solving for

T_0

yields

T_0 = 15

. Thus, the model is

T = 27.5d + 15

. To find the depth when the temperature is

114^\circ\text{C}

, substitute

T = 114

into the model:

114 = 27.5d + 15

. Subtracting

15

from both sides gives

99 = 27.5d

, and dividing by

27.5

yields

d = 3.6

.

Step-by-Step Solution

1

Find the rate of change (slope) of the temperature with respect to depth.

The slope is

27.5^\circ\text{C/km}

.

The temperature increases by

22^\circ\text{C}

for every

0.8\text{ km}

of depth, so the rate of change is

\frac{22^\circ\text{C}}{0.8\text{ km}} = 27.5^\circ\text{C/km}

.

2

Set up a linear model and find the surface temperature (y-intercept).

The linear model is

T = 27.5d + 15

.

Using the slope-intercept form

T = md + T_0

and substituting the given values

d = 2.4

and

T = 81

gives

81 = 27.5(2.4) + T_0

, which simplifies to

81 = 66 + T_0

, so

T_0 = 15

.

3

Substitute the target temperature into the linear model and solve for the target depth.

d = 3.6

Substitute

T = 114

into the equation

T = 27.5d + 15

to get

114 = 27.5d + 15

. Subtracting

15

from both sides gives

99 = 27.5d

. Dividing both sides by

27.5

yields

d = 3.6

.

Key Concept

Interpreting slope and solving for values in a linear relationship context.

Alternative Method

Find the required temperature increase:

114^\circ\text{C} - 81^\circ\text{C} = 33^\circ\text{C}

. Set up a proportion using the rate of

22^\circ\text{C}

increase per

0.8\text{ km}

to find the change in depth

\Delta d

:

\frac{33^\circ\text{C}}{\Delta d} = \frac{22^\circ\text{C}}{0.8\text{ km}} \implies \Delta d = \frac{33 \times 0.8}{22} = 1.2\text{ km}

. Add this change in depth to the initial depth to find the final depth:

2.4 + 1.2 = 3.6\text{ km}

.

Estimated Time:2m 0s

Question 247Question →

In the system of equations below, $x$ and $y$ are real numbers.

4^x \cdot 8^y = 128

9^x \cdot 3^y = 243

What is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is

2

.

To solve the system of equations, express all terms in each equation using common bases. In the first equation,

4^x \cdot 8^y = 128

can be written as

(2^2)^x \cdot (2^3)^y = 2^7

, which simplifies to

2^{2x+3y} = 2^7

, meaning

2x + 3y = 7

. In the second equation,

9^x \cdot 3^y = 243

can be written as

(3^2)^x \cdot 3^y = 3^5

, which simplifies to

3^{2x+y} = 3^5

, meaning

2x + y = 5

. Subtracting the two equations gives

(2x + 3y) - (2x + y) = 7 - 5 \implies 2y = 2 \implies y = 1

. Substituting

y = 1

back into

2x + y = 5

gives

2x + 1 = 5 \implies 2x = 4 \implies x = 2

.

Step-by-Step Solution

1

Rewrite the first equation using a common base of 2.

2x + 3y = 7

By converting

4^x

to

(2^2)^x = 2^{2x}

,

8^y

to

(2^3)^y = 2^{3y}

, and

128

to

2^7

, we can equate the exponents:

2x + 3y = 7

.

2

Rewrite the second equation using a common base of 3.

2x + y = 5

By converting

9^x

to

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

, keeping

3^y

, and converting

243

to

3^5

, we can equate the exponents:

2x + y = 5

.

3

Solve the system of linear equations for

y

.

y = 1

Subtracting

2x + y = 5

from

2x + 3y = 7

eliminates

x

, leaving

2y = 2

, which gives

y = 1

.

4

Substitute

y = 1

into one of the linear equations to solve for

x

.

x = 2

Substituting

y = 1

into

2x + y = 5

yields

2x + 1 = 5

, which simplifies to

2x = 4

, so

x = 2

.

Key Concept

Solving systems of exponential equations by converting to a common base and applying exponent laws.

Question 248Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + bx + c$ has a vertex at $(h, k)$ . The graph of $f$ is translated $4$ units to the right and $5$ units up to form the graph of a quadratic function $g$ . If the graph of $g$ passes through the point $(1, 16)$ and has a $y$ -intercept of $13$ , what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

The value of

k

is

12

.

By writing the function in vertex form as

f(x) = -(x-h)^2 + k

, we can apply the transformations directly. Translating the graph

4

units to the right and

5

units up gives the function

g(x) = f(x-4) + 5 = -(x-h-4)^2 + k+5

. Using the given points

g(0) = 13

and

g(1) = 16

, we set up a system of equations:

-(h+4)^2 + k+5 = 13

and

-(h+3)^2 + k+5 = 16

. Solving these simultaneously gives

h = -2

and

k = 12

.

Step-by-Step Solution

1

Write the function

f(x)

in vertex form.

f(x) = -(x-h)^2 + k

Since the leading coefficient of

f(x) = -x^2 + bx + c

is

-1

, the vertex form must have a coefficient of

-1

.

2

Represent the translated function

g(x)

.

g(x) = -(x-h-4)^2 + k+5

Translating a function

f(x)

by

4

units to the right and

5

units up yields

g(x) = f(x-4) + 5

.

3

Use the

y

-intercept of

g(x)

to write the first equation.

k - (h+4)^2 = 8

The

y

-intercept is the value of the function at

x = 0

, so

g(0) = 13

.

4

Use the point

(1, 16)

on

g(x)

to write the second equation.

k - (h+3)^2 = 11

The graph passing through

(1, 16)

means

g(1) = 16

.

5

Solve the system of equations for

h

and

k

.

h = -2

and

k = 12

Equating the expressions for

k

gives

(h+4)^2 + 8 = (h+3)^2 + 11

, which simplifies to

2h = -4

, so

h = -2

. Substituting

h = -2

into

k = (h+4)^2 + 8

yields

k = 12

.

Key Concept

Translating quadratic functions and solving systems of quadratic equations.

Question 249Question →

In the inequality $3(4 - 2x) - 5(x - c) \geq -7$ , where $c$ is a constant, the solution set consists of all values of $x$ such that $x \leq 3$ . What is the value of $c$ ?

Show answer & explanation

Answer: 2.8

Answer

2.8

To find the value of

c

, first simplify the inequality

3(4 - 2x) - 5(x - c) \geq -7

by expanding the terms using the distributive property, which yields

12 - 6x - 5x + 5c \geq -7

. Combining like terms gives

-11x + 12 + 5c \geq -7

. Next, isolate the variable term by subtracting

12

and

5c

from both sides to obtain

-11x \geq -19 - 5c

. Dividing both sides of the inequality by

-11

and reversing the inequality sign results in

x \leq \frac{19 + 5c}{11}

. Given that the solution set consists of all values of

x

such that

x \leq 3

, the boundary value

\frac{19 + 5c}{11}

must equal

3

. Solving the equation

\frac{19 + 5c}{11} = 3

gives

19 + 5c = 33

, which simplifies to

5c = 14

and results in

c = 2.8

.

Step-by-Step Solution

1

Distribute the constants on the left side of the inequality.

12 - 6x - 5x + 5c \geq -7

Applying the distributive property simplifies the parentheses.

2

Combine the variable terms.

-11x + 12 + 5c \geq -7

Grouping like terms simplifies the inequality.

3

Subtract

12

and

5c

from both sides of the inequality.

-11x \geq -19 - 5c

This isolates the term containing

x

.

4

Divide both sides by

-11

and reverse the direction of the inequality.

x \leq \frac{19 + 5c}{11}

Dividing by a negative number requires flipping the inequality sign to maintain equivalence.

5

Set the algebraic boundary

\frac{19 + 5c}{11}

equal to the given boundary value of

3

and solve for

c

.

c = 2.8

Since the solution set is

x \leq 3

, the boundary values must be equivalent.

Key Concept

Solving multi-step linear inequalities in one variable containing parameters.

Question 250Question →

The graph of the equation $ax + by = 24$ , where $a$ and $b$ are constants, is a line in the $xy$ -plane. If this line passes through the points $(2, 9)$ and $(6, 3)$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is

5

.

Substituting the coordinates of the two points

(2, 9)

and

(6, 3)

into the given linear equation

ax + by = 24

yields a system of two equations:

2a + 9b = 24

and

6a + 3b = 24

. Simplifying the second equation gives

2a + b = 8

, which means

b = 8 - 2a

. Substituting this expression for

b

into the first equation yields

2a + 9(8 - 2a) = 24

, which simplifies to

-16a = -48

, or

a = 3

. Plugging

a = 3

back into

b = 8 - 2a

gives

b = 2

. Thus, the value of

a + b

is

3 + 2 = 5

.

Step-by-Step Solution

1

Substitute the point

(2, 9)

into the equation

ax + by = 24

.

2a + 9b = 24

Since the line passes through the point

(2, 9)

, the coordinates must satisfy the equation of the line.

2

Substitute the point

(6, 3)

into the equation

ax + by = 24

.

6a + 3b = 24

Since the line passes through the point

(6, 3)

, the coordinates must satisfy the equation of the line.

3

Solve the system of equations for

a

and

b

.

a = 3

and

b = 2

To find the values of the constants

a

and

b

, we solve the linear system: (1)

2a + 9b = 24

and (2)

6a + 3b = 24

. Dividing the second equation by 3 gives

2a + b = 8

, or

b = 8 - 2a

. Substituting this into the first equation gives

2a + 9(8 - 2a) = 24 \implies 2a + 72 - 18a = 24 \implies -16a = -48 \implies a = 3

. Then,

b = 8 - 2(3) = 2

.

4

Add the values of

a

and

b

.

a + b = 5

The question asks for the value of

a + b

.

Key Concept

Solving systems of linear equations derived from coordinate substitution in a two-variable linear equation.

Estimated Time:1m 30s

Question 251Question →

A system of equations consists of the quadratic function $f(x) = -x^2 + 6x - 2$ and the linear function $g(x) = 2x + 1$ . If the graphs of $f$ and $g$ intersect at the points $(x_1, y_1)$ and $(x_2, y_2)$ in the $xy$ -plane, what is the value of $y_1 + y_2$ ?

Show answer & explanation

Answer: 10

Answer

10

By setting the two equations equal to each other, we obtain

-x^2 + 6x - 2 = 2x + 1

. Rearranging terms to one side gives

x^2 - 4x + 3 = 0

, which factors as

(x-1)(x-3) = 0

. The x-coordinates of the intersection points are

x = 1

and

x = 3

. Substituting these into the linear equation gives the y-coordinates: for

x = 1

,

y = 2(1) + 1 = 3

; for

x = 3

,

y = 2(3) + 1 = 7

. Summing these y-coordinates gives

3 + 7 = 10

.

Step-by-Step Solution

1

Set the quadratic function equal to the linear function to find the x-coordinates of the intersection points.

-x^2 + 6x - 2 = 2x + 1

At the points of intersection, the y-values of both functions must be equal.

2

Rearrange the equation into standard quadratic form and solve for

x

by factoring.

x^2 - 4x + 3 = 0 \Rightarrow (x - 1)(x - 3) = 0 \Rightarrow x = 1 \text{ or } x = 3

Factoring the quadratic equation gives the x-coordinates of the intersection points.

3

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

For

x_1 = 1

,

y_1 = 2(1) + 1 = 3

. For

x_2 = 3

,

y_2 = 2(3) + 1 = 7

.

The linear equation

g(x) = 2x + 1

is simpler to use for finding the y-coordinates than the quadratic equation.

4

Sum the two y-coordinates.

y_1 + y_2 = 3 + 7 = 10

The question asks for the sum of the y-coordinates of the intersection points.

Key Concept

Solving a system consisting of a linear equation and a quadratic equation by substitution or equating them.

Question 252Question →

The quadratic function $f$ is defined by $f(x) = 3(x - 2)^2 + 5$ . In the $xy$ -plane, the graph of $y = f(x)$ intersects the $y$ -axis at the point $(0, c)$ . What is the value of $c$ ?

Show answer & explanation

Answer: 17

Answer

The correct answer is 17.

To find the

y

-intercept of the graph of

y = f(x)

, we substitute

x = 0

into the equation:

f(0) = 3(0 - 2)^2 + 5

. Simplifying the expression inside the parentheses gives

3(-2)^2 + 5

. Squaring

-2

gives

4

, making the expression

3(4) + 5 = 12 + 5 = 17

. Therefore, the graph intersects the

y

-axis at the point

(0, 17)

, which means the value of

c

is 17.

Step-by-Step Solution

1

Set

x = 0

to find the point where the graph intersects the

y

-axis.

f(0) = 3(0 - 2)^2 + 5

The

y

-intercept of any graph in the

xy

-plane occurs when the

x

-coordinate is equal to

0

.

2

Perform the arithmetic evaluation to find the value of

f(0)

.

f(0) = 17

First, evaluate the term inside the parentheses:

0 - 2 = -2

. Squaring

-2

gives

4

. Then, multiply

3

by

4

to get

12

. Finally, add

5

to obtain

17

.

Key Concept

Evaluating a quadratic function at

x = 0

to determine its

y

-intercept.

Question 253Question →

If $z$ is a positive number such that $z^2 - 4z - 12 = 0$ , what is the value of $z$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

To find the positive value of

z

that satisfies the equation

z^2 - 4z - 12 = 0

, we factor the quadratic expression. Finding two numbers that multiply to

-12

and add to

-4

gives

-6

and

2

. Thus, the equation can be factored as

(z - 6)(z + 2) = 0

. Setting each factor to zero yields the solutions

z = 6

and

z = -2

. Since the problem states that

z

is a positive number, we discard the negative solution, leaving

6

as the final answer.

Step-by-Step Solution

1

Factor the quadratic equation

z^2 - 4z - 12 = 0

.

(z - 6)(z + 2) = 0

To solve the quadratic equation by finding two numbers that multiply to -12 and add to -4.

2

Solve for the roots of the equation.

z = 6

or

z = -2

Set each factor equal to zero and solve for

z

.

3

Apply the constraint that

z

must be positive.

z = 6

Discard the negative solution

z = -2

as the problem specifies

z

must be a positive number.

Key Concept

Solving quadratic equations by factoring

Question 254Question →

A certain substance decays radioactively such that the mass of the substance, in grams, remaining after $t$ days is modeled by the function $M(t) = 80 \cdot 2^{-\frac{t}{5}}$ . After how many days will the mass of the substance be $10$ grams?

Show answer & explanation

Answer: 15

Answer

15

To find the number of days after which the mass of the substance is

10

grams, we substitute

10

for

M(t)

in the given function, yielding

10 = 80 \cdot 2^{-\frac{t}{5}}

. Dividing both sides by

80

gives

\frac{10}{80} = 2^{-\frac{t}{5}}

, which simplifies to

\frac{1}{8} = 2^{-\frac{t}{5}}

. Since

\frac{1}{8}

can be written as

2^{-3}

, the equation becomes

2^{-3} = 2^{-\frac{t}{5}}

. Because the bases are the same, we equate the exponents:

-3 = -\frac{t}{5}

. Multiplying both sides by

-5

gives

t = 15

.

Step-by-Step Solution

1

Set the mass

M(t)

equal to

10

in the given function.

10 = 80 \cdot 2^{-\frac{t}{5}}

We want to find the value of

t

when the remaining mass of the substance is

10

grams.

2

Divide both sides of the equation by

80

.

\frac{1}{8} = 2^{-\frac{t}{5}}

To isolate the exponential expression.

3

Express both sides of the equation with a common base of

2

.

2^{-3} = 2^{-\frac{t}{5}}

Since

8 = 2^3

, the fraction

\frac{1}{8}

can be written as

2^{-3}

. Having the same base on both sides allows us to equate the exponents.

4

Set the exponents equal to each other and solve for

t

.

t = 15

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

-3 = -\frac{t}{5}

.

Key Concept

Solving exponential equations by finding a common base.

Question 255Question →

For all $x > 1$ , the expression $\frac{2x^2 + 7x - 4}{x^2 - 1} \cdot \frac{x - 1}{2x - 1}$ is equivalent to $\frac{x+k}{x+1}$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the constant

k

is 4.

Factoring the numerator

2x^2 + 7x - 4

yields

(2x - 1)(x + 4)

and factoring the denominator

x^2 - 1

yields

(x - 1)(x + 1)

. Substituting these factored forms into the given product gives

\frac{(2x - 1)(x + 4)}{(x - 1)(x + 1)} \cdot \frac{x - 1}{2x - 1}

. Canceling the common factors

(2x - 1)

and

(x - 1)

simplifies the expression to

\frac{x + 4}{x + 1}

. Comparing this to

\frac{x + k}{x + 1}

shows that

k = 4

.

Step-by-Step Solution

1

Factor the quadratic expression in the numerator:

2x^2 + 7x - 4

.

(2x - 1)(x + 4)

Factoring the numerator helps identify common factors that can be simplified.

2

Factor the difference of squares in the denominator:

x^2 - 1

.

(x - 1)(x + 1)

Factoring the denominator helps identify common factors that can be simplified.

3

Multiply the rational expressions and cancel out the common factors.

\frac{x + 4}{x + 1}

Since

x > 1

, the terms

(2x - 1)

and

(x - 1)

are not equal to zero and can be canceled.

4

Compare the resulting expression with

\frac{x + k}{x + 1}

to find the value of

k

.

k = 4

By matching the numerators of the equivalent expressions,

x + 4 = x + k

, which gives

k = 4

.

Key Concept

Factoring and simplifying products of rational expressions

Question 256Question →

In the $xy$ -plane, a point with coordinates $(x, y)$ lies in the region defined by the system of inequalities below.

y + 2x \leq 12

x - 2y \leq 6

x \geq 2

What is the maximum possible value of $y$ ?

Show answer & explanation

Answer: 8

Answer

8

The system of inequalities bounds the solution set. The upper boundary is given by

y \leq -2x + 12

. Because the slope is negative, the maximum value of

y

on this boundary occurs at the smallest possible value of

x

. The constraint

x \geq 2

dictates that the minimum value of

x

is 2. Substituting

x = 2

into the boundary equation yields

y = -2(2) + 12 = 8

. Checking this coordinate against the third inequality,

2 - 2(8) = -14 \leq 6

, verifies that

(2, 8)

is a valid solution.

Step-by-Step Solution

1

Express the first two inequalities in terms of y.

y \leq -2x + 12

and

y \geq \frac{1}{2}x - 3

This helps identify the upper and lower boundaries of the solution region.

2

Determine the boundary line that limits the maximum values of y.

The upper boundary is the line

y = -2x + 12

.

Since the inequality is

y \leq -2x + 12

, any solution must lie on or below this line.

3

Find the maximum value of y on this boundary line given the constraint

x \geq 2

.

y \leq -2(2) + 12 = 8

Since the slope of the boundary line is negative, y is maximized when x is at its minimum value, which is 2.

4

Verify that the point

(2, 8)

satisfies the inequality

x - 2y \leq 6

.

2 - 2(8) = -14 \leq 6

, which is true.

This confirms that the point

(2, 8)

is indeed in the solution set of the system.

Key Concept

Maximizing a coordinate value subject to a system of linear inequalities in two variables

Alternative Method

Instead of graphing or checking boundaries, we can algebraically solve for the boundary. Since

x \geq 2

, multiplying by

-2

and reversing the inequality gives

-2x \leq -4

. Adding 12 to both sides gives

-2x + 12 \leq 8

. Since

y \leq -2x + 12

, we get

y \leq 8

. Checking if

y = 8

and

x = 2

satisfies the second inequality

x - 2y \leq 6

confirms that

2 - 16 = -14 \leq 6

is true, meaning

y = 8

is indeed a valid solution and thus the maximum.

Estimated Time:1m 30s

Question 257Question →

A shipping container has a maximum weight capacity of $24,150$ kilograms. The container is loaded with $12$ machinery units, each weighing $1,150$ kilograms. The remaining space will be filled with packing crates, each weighing $180$ kilograms. A safety regulation requires that a clearance weight of at least $15\%$ of the total loaded weight (the combined weight of the machinery units and the packing crates) must be left unused. What is the maximum number of packing crates that can be loaded into the container?

Show answer & explanation

Answer: 40

Answer

The maximum number of packing crates that can be loaded is 40.

By setting up the inequality representing the physical constraints, we find that the number of packing crates,

x

, must satisfy

x \le 40

. Since the question asks for the maximum number of packing crates, the maximum value is 40.

Step-by-Step Solution

1

Calculate the constant weight of the machinery units.

The total weight of the

12

machinery units is

12 \times 1,150 = 13,800

kilograms.

This establishes the base weight that is already loaded in the container.

2

Define the variable and write the expression for the total loaded weight.

Let

x

be the number of packing crates. The total loaded weight is

13,800 + 180x

kilograms.

This represents the combined weight of the machinery and the crates in terms of the variable

x

.

3

Set up the inequality representing the safety clearance requirement.

24,150 - (13,800 + 180x) \ge 0.15(13,800 + 180x)

The unused weight capacity (maximum capacity minus loaded weight) must be at least

15\%

of the loaded weight.

4

Solve the inequality for

x

.

24,150 \ge 1.15(13,800 + 180x) \implies 21,000 \ge 13,800 + 180x \implies 7,200 \ge 180x \implies x \le 40

.

Isolating

x

gives the range of allowable values for the number of packing crates.

Key Concept

Formulating and solving linear inequalities in one variable based on real-world constraints.

Question 258Question →

If $(x - 4)^2 = 81$ and $x > 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: 13

Answer

The value of

x

is 13.

Taking the square root of both sides of

(x - 4)^2 = 81

yields

x - 4 = 9

or

x - 4 = -9

. Solving these linear equations gives

x = 13

or

x = -5

. Since

x > 0

is specified, the correct value of

x

is 13.

Step-by-Step Solution

1

Take the square root of both sides of the equation.

x - 4 = 9

or

x - 4 = -9

Applying the square root property to solve the quadratic equation.

2

Solve each linear equation for

x

.

x = 13

or

x = -5

Adding 4 to both sides of each equation.

3

Apply the given constraint that

x > 0

.

x = 13

Since

-5

is not greater than 0, the only positive solution is 13.

Key Concept

Solving quadratic equations by taking square roots

Question 259Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(3, 11)$ and $(7, 23)$ . The function $g$ is defined by $g(x) = f(2x) - 5$ . What is the value of $g(4)$ ?

Show answer & explanation

Answer: 21

Answer

21

The correct answer is 21. First, find the slope of the linear function

f

using the given points

(3, 11)

and

(7, 23)

:

m = \frac{23 - 11}{7 - 3} = 3

. The equation of the line is

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

. To find

g(4)

, substitute

x = 4

into the definition of

g(x)

:

g(4) = f(2 \cdot 4) - 5 = f(8) - 5

. Evaluating

f(8)

gives

3(8) + 2 = 26

. Finally, subtracting

5

gives

g(4) = 26 - 5 = 21

.

Step-by-Step Solution

1

Determine the equation of the linear function

f(x)

f(x) = 3x + 2

First find the slope

m = \frac{23 - 11}{7 - 3} = 3

. Then, use the point-slope formula with

(3, 11)

to get

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

.

2

Express

g(4)

in terms of

f

g(4) = f(8) - 5

Substitute

x = 4

into the definition

g(x) = f(2x) - 5

to get

g(4) = f(2(4)) - 5

.

3

Calculate the value of

f(8)

and

g(4)

g(4) = 21

Evaluate

f(8) = 3(8) + 2 = 26

, then subtract

5

to obtain

g(4) = 26 - 5 = 21

.

Key Concept

Linear Functions and Graphs

Question 260Question →

A scientist is monitoring the population of two species of bacteria, Species A and Species B, in a controlled environment. Initially, the combined population of the two species is $12,000$ . Over the next week, the population of Species A doubles, and the population of Species B triples. If the total combined population of both species is $29,000$ at the end of the week, what was the initial population of Species A?

Show answer & explanation

Answer: 7000

Answer

The initial population of Species A was 7000.

To find the initial population of Species A, we set up a system of linear equations representing the total initial population and the population after one week. Let

x

represent the initial population of Species A and

y

represent the initial population of Species B. The initial combined population is given by

x + y = 12,000

. After one week, the population of Species A doubles (

2x

) and Species B triples (

3y

), so the new combined population is

2x + 3y = 29,000

. We can express

y

in terms of

x

as

y = 12,000 - x

. Substituting this expression into the second equation gives

2x + 3(12,000 - x) = 29,000

. Distributing and simplifying yields

-x + 36,000 = 29,000

, which simplifies to

x = 7,000

. Therefore, the initial population of Species A was 7,000.

Step-by-Step Solution

1

Define the variables and set up the first equation.

x + y = 12,000

Let

x

be the initial population of Species A and

y

be the initial population of Species B. Their combined initial population is

12,000

.

2

Set up the second equation based on the growth after one week.

2x + 3y = 29,000

The population of Species A doubles to

2x

and Species B triples to

3y

, summing to a total of

29,000

.

3

Solve for

x

using substitution.

x = 7,000

From the first equation,

y = 12,000 - x

. Substituting this into the second equation gives

2x + 3(12,000 - x) = 29,000

. Distributing the

3

yields

2x + 36,000 - 3x = 29,000

. Combining like terms results in

-x + 36,000 = 29,000

. Subtracting

36,000

from both sides gives

-x = -7,000

, which simplifies to

x = 7,000

.

Key Concept

Solving systems of linear equations using substitution or elimination methods

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

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