All practice questions

Question 221Question →

A local gym charges a one-time registration fee plus a constant monthly fee. The total cost, $C$ , in dollars, for a membership of $m$ months is modeled by the equation $C = 35m + 50$ . What is the one-time registration fee, in dollars, for the gym?

Show answer & explanation

Answer: 50

Answer

The one-time registration fee is 50 dollars.

In the linear model

C = 35m + 50

, the constant term

50

represents the value of the function when

m = 0

. In this context,

m = 0

corresponds to

0

months of membership, meaning no monthly fees have been incurred yet. Therefore, the value of

50

represents the initial, one-time registration fee.

Step-by-Step Solution

1

Analyze the linear equation

C = 35m + 50

to identify the slope and the y-intercept.

The slope is

35

(the coefficient of

m

) and the y-intercept is

50

(the constant term).

In a linear equation of the form

y = mx + b

, the constant

b

represents the value of

y

when

x = 0

(the y-intercept), and the coefficient

m

represents the rate of change (the slope).

2

Interpret the meaning of the y-intercept in the given context.

At

m = 0

months, the total cost

C

is

C = 35(0) + 50 = 50

dollars.

The cost at

0

months represents the upfront, one-time fee before any monthly fees are added, which is the registration fee.

Key Concept

Interpreting the y-intercept of a linear function in a real-world context.

Question 222Question →

In the quadratic equation $x^2 - kx + 36 = 0$ , $k$ is a positive constant. If the equation has exactly one real solution, what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

The value of

k

is

12

.

The quadratic equation

x^2 - kx + 36 = 0

has exactly one real solution when its discriminant is equal to zero. The discriminant of

ax^2 + bx + c = 0

is

b^2 - 4ac

. For this equation,

a = 1

,

b = -k

, and

c = 36

. Setting the discriminant to zero gives

(-k)^2 - 4(1)(36) = 0

, which simplifies to

k^2 - 144 = 0

. Solving for

k

gives

k^2 = 144

, so

k = 12

or

k = -12

. Since the problem states that

k

is a positive constant, the correct value of

k

is

12

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

x^2 - kx + 36 = 0

.

a = 1

,

b = -k

, and

c = 36

.

To apply the discriminant formula, we need the values of

a

,

b

, and

c

.

2

Set the discriminant

\Delta = b^2 - 4ac

to

0

.

(-k)^2 - 4(1)(36) = 0

.

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

3

Solve the equation for

k

.

k^2 - 144 = 0 \implies k^2 = 144 \implies k = 12

(since

k

is a positive constant).

Solving the equation yields

k = \pm 12

, and we select the positive value as specified in the problem.

Key Concept

Discriminant of a quadratic equation

Question 223Question →

A parabola in the $xy$ -plane has its vertex at $(3, 8)$ and passes through the point $(5, 16)$ . If the equation of the parabola is written in the form $y = a(x - 3)^2 + 8$ , what is the value of the constant $a$ ?

Show answer & explanation

Answer: 2

Answer

2

Substituting the coordinates

(5, 16)

into

y = a(x - 3)^2 + 8

yields

16 = a(5 - 3)^2 + 8

. Simplifying the term

(5 - 3)^2

gives

4

, leading to the equation

16 = 4a + 8

. Subtracting 8 from both sides gives

8 = 4a

, and dividing by 4 gives the final answer of 2.

Step-by-Step Solution

1

Substitute the point

(5, 16)

into the equation.

16 = a(5 - 3)^2 + 8

Since the parabola passes through the point

(5, 16)

, these coordinates must satisfy the equation.

2

Simplify the squared term.

16 = 4a + 8

Subtract 3 from 5 to get 2, and square 2 to get 4.

3

Solve for the constant

a

.

a = 2

Subtract 8 from both sides to get

8 = 4a

, then divide by 4 to find

a

.

Key Concept

Determining a constant in a quadratic function given its vertex form and a point on the graph.

Question 224Question →

An online service provider offers two monthly subscription plans. Under Plan A, the customer pays a flat monthly fee of $C$ dollars. Under Plan B, the monthly cost, in dollars, is determined by the expression $1.5(20 - x) - 0.8(3x - 5)$ , where $x$ is the number of premium features the customer uses. The provider wants Plan B to be strictly cheaper than Plan A for any customer who uses more than 4 premium features. If $C$ is an integer, what is the minimum possible value of $C$ ?

Show answer & explanation

Answer: 19

Answer

19

To find the minimum integer value of

C

, we simplify Plan B's cost expression to

34 - 3.9x

and set up the inequality

34 - 3.9x < C

. Solving for

x

by dividing by

-3.9

and reversing the inequality sign gives

x > \frac{34 - C}{3.9}

. For Plan B to be cheaper than Plan A for all customers using more than 4 features, the solution set

x > \frac{34 - C}{3.9}

must contain the interval

x > 4

. This requires the boundary point to be at most 4, so

\frac{34 - C}{3.9} \le 4

. Solving this inequality yields

C \ge 18.4

. The smallest integer value greater than or equal to

18.4

is 19.

Step-by-Step Solution

1

Simplify the cost expression for Plan B

34 - 3.9x

To combine like terms and express Plan B's cost in standard linear form.

2

Set up the inequality stating Plan B is strictly cheaper than Plan A

34 - 3.9x < C

Plan B is cheaper than Plan A when its cost is less than

C

dollars.

3

Solve the inequality for

x

in terms of

C

x > \frac{34 - C}{3.9}

Isolating

x

allows us to find the threshold number of premium features, remembering to reverse the inequality direction when dividing by the negative coefficient

-3.9

.

4

Relate the threshold condition to the given minimum number of premium features

\frac{34 - C}{3.9} \le 4

For Plan B to be cheaper for any

x > 4

, the solution interval

x > \frac{34 - C}{3.9}

must cover the entire interval

x > 4

. Thus, the boundary point must be at most 4.

5

Solve the boundary inequality for

C

C \ge 18.4

Multiplying by

3.9

and isolating

C

gives the lower bound for the cost of Plan A.

6

Find the minimum integer value for

C

19

Since

C

must be an integer and at least

18.4

, the smallest integer that satisfies this inequality is 19.

Key Concept

Solving linear inequalities in one variable with parameter constraints and real-world conditions.

Question 225Question →

If $25^a = 125^b$ , where $a$ and $b$ are positive constants, what is the value of $\frac{a}{b}$ ?

Show answer & explanation

Answer: 1.5

Answer

1.5

To find the value of

\frac{a}{b}

, we can rewrite the bases of the equation

25^a = 125^b

using a common base of 5:

(5^2)^a = (5^3)^b

. Applying the exponent power rule gives

5^{2a} = 5^{3b}

. Since the bases are identical, their exponents must be equal, so

2a = 3b

. Dividing both sides of this equation by

2b

yields the ratio

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

, which is

1.5

.

Step-by-Step Solution

1

Express the bases as powers of 5

(5^2)^a = (5^3)^b

To solve the equation, express both sides with a common base of 5 since

25 = 5^2

and

125 = 5^3

.

2

Apply the power rule of exponents

5^{2a} = 5^{3b}

According to the exponent rules,

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

.

3

Set the exponents equal to each other

2a = 3b

Because the bases on both sides of the equation are equal and positive, their exponents must also be equal.

4

Solve for the ratio

\frac{a}{b} = 1.5

Divide both sides of the equation

2a = 3b

by

2b

to isolate the ratio

\frac{a}{b}

.

Key Concept

Solving exponential equations by expressing bases in terms of a common base and equating exponents.

Question 226Question →

In the quadratic equation $8x^2 - kx + 27 = 0$ , where $k$ is a positive constant, one of the roots is the square of the other root. What is the value of $k$ ?

Show answer & explanation

Answer: 30

Answer

30

For the quadratic equation

8x^2 - kx + 27 = 0

, Vieta's formulas state that the product of the roots is

\frac{27}{8}

and the sum of the roots is

\frac{k}{8}

. Letting the roots be

r

and

r^2

, we obtain the equation

r^3 = \frac{27}{8}

, which gives

r = \frac{3}{2}

. The other root is

r^2 = \frac{9}{4}

. Summing these roots gives

\frac{3}{2} + \frac{9}{4} = \frac{15}{4}

. Equating this sum to the formula for the sum of the roots yields

\frac{k}{8} = \frac{15}{4}

, which simplifies to

k = 30

.

Step-by-Step Solution

1

Determine the product of the roots using Vieta's formulas.

The product of the roots is

r_1 r_2 = \frac{27}{8}

.

For any quadratic equation

ax^2 + bx + c = 0

, the product of its roots is given by

\frac{c}{a}

.

2

Use the root relationship to solve for one of the roots.

r_2^3 = \frac{27}{8} \implies r_2 = \frac{3}{2}

.

Substituting

r_1 = r_2^2

into the product equation yields

r_2^3 = \frac{27}{8}

.

3

Determine the value of the second root.

r_1 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}

.

The first root is the square of the second root.

4

Calculate the sum of the two roots.

r_1 + r_2 = \frac{9}{4} + \frac{3}{2} = \frac{15}{4}

.

The sum of the roots is needed to relate back to the linear coefficient of the quadratic equation.

5

Solve for the constant

k

using the sum of the roots.

\frac{k}{8} = \frac{15}{4} \implies k = 30

.

By Vieta's formulas, the sum of the roots is

-\frac{b}{a} = \frac{k}{8}

.

Key Concept

Quadratic Equations and Vieta's Formulas

Question 227Question →

An athlete participates in a charity event where they earn money by swimming laps in a pool and running laps on a track. The athlete earns $s$ dollars for each lap swum and $r$ dollars for each lap run. Completing 10 laps in the pool and 15 laps on the track raises a total of $135. Completing 12 laps in the pool and 20 laps on the track raises a total of$ 176. What is the value of $r$ ?

Show answer & explanation

Answer: 7

Answer

The value of

r

is 7.

The correct answer is 7. To find this, we translate the word problem into a system of two linear equations:

10s + 15r = 135

and

12s + 20r = 176

. Simplifying these equations by dividing by their common factors yields

2s + 3r = 27

and

3s + 5r = 44

. Multiplying the first equation by 3 gives

6s + 9r = 81

, and multiplying the second equation by 2 gives

6s + 10r = 88

. Subtracting the two equations eliminates

s

and gives

r = 7

.

Step-by-Step Solution

1

Write the system of linear equations representing the athlete's earnings.

The system is:

10s + 15r = 135

12s + 20r = 176

To represent the relationships between the number of laps completed and the total amount raised.

2

Simplify both equations to work with smaller coefficients.

Divide the first equation by 5:

2s + 3r = 27

Divide the second equation by 4:

3s + 5r = 44

Simplifying equations reduces the likelihood of arithmetic errors.

3

Eliminate the variable

s

to solve for

r

.

Multiply the first simplified equation by 3 and the second by 2:

3(2s + 3r) = 3(27) \implies 6s + 9r = 81

2(3s + 5r) = 2(44) \implies 6s + 10r = 88

Subtract the first equation from the second:

(6s + 10r) - (6s + 9r) = 88 - 81 \implies r = 7

Using the elimination method allows us to solve directly for the target variable,

r

.

Key Concept

Solving a system of linear equations in two variables using elimination or substitution.

Question 228Question →

The system of equations below consists of a linear equation and a quadratic equation:

\begin{aligned} y - 2x &= 5 \\ y &= x^2 - 3x - 1 \end{aligned}

Let

(x_1, y_1)

and

(x_2, y_2)

represent the two distinct real coordinate points where the graphs of these equations intersect. If

y_1 > y_2

, what is the value of

x_1 - x_2

?

Show answer & explanation

Answer: 7

Answer

7

Solving the system algebraically by expressing

y

as

2x + 5

and substituting it into the quadratic equation leads to

x^2 - 5x - 6 = 0

. Factoring this equation yields

x = 6

and

x = -1

. Evaluating these values in the linear equation gives the coordinates

(6, 17)

and

(-1, 3)

. Since

17 > 3

, the point

(6, 17)

corresponds to

(x_1, y_1)

and

(-1, 3)

corresponds to

(x_2, y_2)

. The difference

x_1 - x_2

is equal to

6 - (-1) = 7

.

Step-by-Step Solution

1

Solve for

y

in the linear equation to express it in terms of

x

.

y = 2x + 5

This allows for substitution into the quadratic equation to eliminate one variable.

2

Substitute

2x + 5

for

y

in the quadratic equation.

2x + 5 = x^2 - 3x - 1

To create a single-variable quadratic equation in terms of

x

.

3

Rearrange the equation by subtracting

2x

and

5

from both sides to set it equal to zero.

x^2 - 5x - 6 = 0

Setting the quadratic equation to zero is the standard first step to find its roots.

4

Factor the quadratic trinomial.

(x - 6)(x + 1) = 0

, which gives

x = 6

and

x = -1

.

Factoring finds the

x

-coordinates of the points of intersection.

5

Substitute the

x

-values back into the linear equation

y = 2x + 5

to determine the corresponding

y

-values.

For

x = 6

,

y = 2(6) + 5 = 17

. For

x = -1

,

y = 2(-1) + 5 = 3

. The intersection points are

(6, 17)

and

(-1, 3)

.

This defines the full coordinate pairs of the system's solutions.

6

Apply the condition

y_1 > y_2

to identify

(x_1, y_1)

and

(x_2, y_2)

.

Since

17 > 3

, the first solution is

(x_1, y_1) = (6, 17)

and the second solution is

(x_2, y_2) = (-1, 3)

.

This aligns the coordinate values with the variables defined in the problem constraint.

7

Calculate the value of

x_1 - x_2

.

6 - (-1) = 7

Subtracting a negative value is equivalent to adding its absolute value, yielding the final required answer.

Key Concept

Nonlinear Systems of Equations

Question 229Question →

In the system of equations below,

k

is a constant.

\begin{aligned} x^2 + y^2 + 8y &= 9 \\ x^2 + y &= k \end{aligned}

If the system has exactly 3 distinct real solutions

(x, y)

, what is the value of

k

?

Show answer & explanation

Answer: 1

Answer

The value of

k

is 1.

Substituting

x^2 = k - y

into

x^2 + y^2 + 8y = 9

gives

y^2 + 7y + k - 9 = 0

. For the system to have exactly 3 distinct real solutions, the vertex of the parabola

y = -x^2 + k

must lie on the circle, which corresponds to the root

y = k

. Substituting

y = k

into the quadratic equation yields

k^2 + 8k - 9 = 0

, which gives

k = 1

or

k = -9

. For

k = 1

, the roots of the quadratic are

y = 1

and

y = -8

. The root

y = 1

yields 1 real solution,

(0, 1)

, and the root

y = -8

yields 2 real solutions,

(3, -8)

and

(-3, -8)

, for a total of 3 real solutions. For

k = -9

, the roots are

y = -9

and

y = 2

. The root

y = 2

does not yield any real solutions for

x

because

2 > -9

, so the system has only 1 real solution. Thus,

k = 1

.

Step-by-Step Solution

1

Express

x^2

in terms of

y

and

k

from the second equation.

x^2 = k - y

(with constraint

y \le k

for real

x

)

To prepare for substitution into the first equation and establish the boundary condition for real solutions.

2

Substitute

x^2 = k - y

into the first equation.

y^2 + 7y + k - 9 = 0

To create a single quadratic equation in terms of

y

.

3

Analyze the conditions on the roots of the quadratic equation to get exactly 3 distinct real solutions.

One root must equal

k

and the other must be less than

k

.

A root

y = k

yields 1 real solution for

x

(

x = 0

), while a root

y < k

yields 2 real solutions (

x = \pm\sqrt{k-y}

).

4

Find the candidate values of

k

by setting

y = k

in the quadratic equation.

k^2 + 8k - 9 = 0 \implies k = 1

or

k = -9

To find the values of

k

where the parabola's vertex lies on the circle.

5

Verify which candidate value of

k

satisfies all conditions.

For

k = 1

, the roots are

y = 1

and

y = -8 < 1

(3 solutions). For

k = -9

, the roots are

y = -9

and

y = 2 > -9

(1 solution). Therefore,

k = 1

.

To ensure the second root is strictly less than

k

, guaranteeing exactly 3 solutions.

Key Concept

Analyzing the number of solutions in a nonlinear system of equations using algebraic substitution and boundary constraints.

Estimated Time:3m 0s

Question 230Question →

An online retailer uses a linear relationship to determine the shipping cost of an order based on the total weight of the items in the order. The table below shows the shipping cost, $y$ , in dollars, for an order with a total weight of $x$ pounds.

Weight ( $x$ pounds)	Shipping Cost ( $y$ dollars)
$2$	$9.50$
$6$	$17.50$

What is the shipping cost, in dollars, for an order with a total weight of $11$ pounds?

Show answer & explanation

Answer: 27.5

Answer

The correct shipping cost for an order weighing 11 pounds is 27.5 dollars.

The linear relationship has a constant rate of change (slope). Using the coordinates from the table,

(2, 9.50)

and

(6, 17.50)

, the slope

m

is calculated as

\frac{17.50 - 9.50}{6 - 2} = \frac{8}{4} = 2

. Using the point-slope form, the equation of the line is

y - 9.50 = 2(x - 2)

, which simplifies to

y = 2x + 5.50

. Substituting

x = 11

gives

y = 2(11) + 5.50 = 27.5

.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the linear relationship.

m = 2

Determines the rate at which the shipping cost increases per additional pound of weight.

2

Formulate the linear equation representing the relationship.

y = 2x + 5.50

Establishes a functional relationship to calculate costs for any given weight.

3

Evaluate the function at the target weight of 11 pounds.

y = 27.5

Determines the final cost of shipping for the specified weight of 11 pounds.

Key Concept

Determining and evaluating linear functions represented in tabular form

Estimated Time:1m 30s

Question 231Question →

The function $f$ is defined by $f(x) = a \cdot 3^x$ , where $a$ is a constant. If $f(2) = 45$ , what is the value of $f(1)$ ?

Show answer & explanation

Answer: 15

Answer

15

To find the value of

f(1)

, we first determine the value of the constant

a

. We are given that

f(2) = 45

, so substituting

x = 2

into the function definition

f(x) = a \cdot 3^x

gives

45 = a \cdot 3^2

. Simplifying

3^2

to

9

yields

45 = 9a

, which means

a = 5

. Now that we know

a = 5

, we can write the function as

f(x) = 5 \cdot 3^x

. To find

f(1)

, we substitute

x = 1

into this equation, yielding

f(1) = 5 \cdot 3^1 = 15

.

Step-by-Step Solution

1

Substitute the given point

(2, 45)

into the function equation to solve for

a

.

a = 5

Since

f(2) = 45

, we have

45 = a \cdot 3^2 = 9a

, which gives

a = 5

.

2

Evaluate the function at

x = 1

using the value of

a = 5

.

f(1) = 15

Substituting

a = 5

and

x = 1

into

f(x) = a \cdot 3^x

gives

f(1) = 5 \cdot 3^1 = 15

.

Key Concept

Evaluating and solving exponential functions given initial conditions or points.

Question 232Question →

A deep-sea research submersible's internal cabin pressure, $P$ , in atmospheres ( $\text{atm}$ ), is modeled by a linear function of its depth below the ocean surface, $d$ , in meters. At the surface ( $d = 0$ ), the internal pressure is $1.0\text{ atm}$ . For every increase in depth of $100\text{ meters}$ , the internal pressure increases by $0.05\text{ atm}$ . The submersible descends from the surface at a constant rate of $2.5\text{ meters per second}$ . What is the rate of increase of the internal cabin pressure, in $\text{atm}$ per hour, as the submersible descends?

Show answer & explanation

Answer: 4.5

Answer

The rate of increase of the internal cabin pressure is

4.5

atmospheres per hour.

The rate of change of internal pressure with depth is

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

. Since the submersible descends at a rate of

2.5\text{ m/s}

, we convert this rate of depth change to hours:

2.5\text{ m/s} \times 3600\text{ s/hr} = 9000\text{ m/hr}

. Multiplying the pressure change per meter by the depth change per hour gives the rate of change of the internal pressure per hour:

0.0005\text{ atm/m} \times 9000\text{ m/hr} = 4.5\text{ atm/hr}

.

Step-by-Step Solution

1

Calculate the rate of pressure increase per meter of depth.

0.0005\text{ atm/m}

The pressure increases by

0.05\text{ atm}

for every

100\text{ meters}

of depth, giving a rate of change of

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

.

2

Determine the distance descended by the submersible in one hour.

9000\text{ meters}

With

3600\text{ seconds}

in one hour and a descent speed of

2.5\text{ m/s}

, the submersible descends a total of

2.5\text{ m/s} \times 3600\text{ s} = 9000\text{ meters}

in one hour.

3

Calculate the rate of internal pressure increase per hour.

4.5\text{ atm/hr}

Multiply the rate of change of pressure per meter (

0.0005\text{ atm/m}

) by the hourly descent distance (

9000\text{ m}

) to obtain

0.0005 \times 9000 = 4.5\text{ atm/hr}

.

Key Concept

Interpreting Linear Relationships in Context

Alternative Method

Write the cabin pressure as a function of time

t

, in seconds:

P(t) = 1.0 + 0.0005(2.5t) = 1.0 + 0.00125t

. The slope of this line,

0.00125\text{ atm/s}

, represents the rate of increase per second. To convert this rate to hours, multiply by

3600\text{ seconds/hour}

:

0.00125 \times 3600 = 4.5\text{ atm/hr}

.

Estimated Time:2m 0s

Question 233Question →

In the $xy$ -plane, the graph of the linear equation $3x + 4y = k$ , where $k$ is a positive constant, is tangent to the circle with equation $x^2 + y^2 - 2x - 4y = 20$ . What is the value of $k$ ?

Show answer & explanation

Answer: 36

Answer

The value of the constant

k

is

36

.

To find the value of

k

where the line

3x + 4y = k

is tangent to the circle, we first rewrite the circle's equation

x^2 + y^2 - 2x - 4y = 20

in standard form by completing the square:

(x^2 - 2x + 1) + (y^2 - 4y + 4) = 20 + 1 + 4

, which simplifies to

(x - 1)^2 + (y - 2)^2 = 25

. This is a circle centered at

(1, 2)

with a radius of

5

. A line is tangent to a circle if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle. Using the distance formula

\frac{|Ax_0 + By_0 - C|}{\sqrt{A^2 + B^2}}

for the point

(1, 2)

and the line

3x + 4y - k = 0

, we set up the equation:

\frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = 5

. This simplifies to

\frac{|11 - k|}{5} = 5

, or

|11 - k| = 25

. Solving the absolute value equation gives

11 - k = 25 \Rightarrow k = -14

and

11 - k = -25 \Rightarrow k = 36

. Since the problem specifies that

k

is a positive constant, the correct answer is

36

.

Step-by-Step Solution

1

Rewrite the circle's equation by completing the square for both variables.

(x - 1)^2 + (y - 2)^2 = 25

To identify the center and radius of the circle.

2

Identify the center and radius from the standard form of the circle's equation.

Center is

(1, 2)

and radius is

5

.

The standard form of a circle is

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

.

3

Apply the condition for tangency: the perpendicular distance from the center

(1, 2)

to the line

3x + 4y - k = 0

must equal the radius

5

.

\frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = 5

A line is tangent to a circle if and only if the distance from the center to the line equals the radius.

4

Simplify the distance equation and solve the resulting absolute value equation for

k

.

|11 - k| = 25

, yielding

k = -14

or

k = 36

.

To find all mathematically possible values of the constant

k

.

5

Select the positive value for

k

as specified by the problem constraints.

k = 36

The problem states that

k

is a positive constant.

Key Concept

The relationship between a line and a circle in a nonlinear system, specifically using the distance from the center to a tangent line to solve for an unknown constant.

Question 234Question →

During a chemistry experiment, the temperature of a liquid increases at a constant rate. The temperature of the liquid is $24^\circ\text{C}$ at $5$ minutes after the experiment starts, and $48^\circ\text{C}$ at $15$ minutes after the experiment starts. The temperature $T$ , in degrees Celsius, of the liquid $t$ minutes after the experiment starts can be modeled by the equation $T = mt + b$ , where $m$ and $b$ are constants. What is the value of $b$ ?

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. By translating the given information into two points on the line,

(5, 24)

and

(15, 48)

, we can find the slope

m = \frac{48 - 24}{15 - 5} = 2.4

. Substituting this slope and the point

(5, 24)

into

T = mt + b

yields

24 = 2.4(5) + b

, which simplifies to

24 = 12 + b

. Solving for

b

gives

b = 12

.

Step-by-Step Solution

1

Identify the data points representing the relationship between time and temperature.

The two data points are

(t_1, T_1) = (5, 24)

and

(t_2, T_2) = (15, 48)

.

These points represent coordinates

(t, T)

on the line representing the temperature over time.

2

Calculate the slope

m

of the linear equation.

m = \frac{48 - 24}{15 - 5} = \frac{24}{10} = 2.4

The slope represents the constant rate of temperature increase per minute.

3

Substitute the slope

m

and the coordinates of one point into the equation

T = mt + b

to solve for the y-intercept

b

.

24 = 2.4(5) + b \implies 24 = 12 + b \implies b = 12

This determines the value of the constant

b

, which corresponds to the initial temperature of the liquid.

Key Concept

Determining the y-intercept of a linear model in two variables given two points.

Question 235Question →

If $2^{2x+2} - 2^{2x} = 12$ for some real number $x$ , what is the value of $2^{4x}$ ?

Show answer & explanation

Answer: 16

Answer

16

By applying the product rule of exponents, the expression

2^{2x+2}

can be rewritten as

2^{2x} \cdot 2^2

, or

4 \cdot 2^{2x}

. Substituting this into the given equation yields

4 \cdot 2^{2x} - 2^{2x} = 12

. Factoring out

2^{2x}

gives

2^{2x}(4 - 1) = 12

, which simplifies to

3 \cdot 2^{2x} = 12

. Dividing both sides of the equation by 3 results in

2^{2x} = 4

. Since

2^{4x}

can be written as

(2^{2x})^2

, substituting 4 for

2^{2x}

gives

4^2 = 16

. Alternatively, solving

2^{2x} = 4

gives

2x = 2

, which means

x = 1

. Substituting

x = 1

into

2^{4x}

yields

2^{4(1)} = 2^4 = 16

.

Step-by-Step Solution

1

Rewrite

2^{2x+2}

using exponent properties.

2^{2x+2} = 2^{2x} \cdot 2^2 = 4 \cdot 2^{2x}

To express both exponential terms with the same base power, allowing them to be factored or combined.

2

Substitute this back into the equation and factor out the common term

2^{2x}

.

2^{2x}(4 - 1) = 12 \implies 3 \cdot 2^{2x} = 12

To isolate the exponential expression.

3

Solve for

2^{2x}

by dividing both sides of the equation by 3.

2^{2x} = 4

To find the value of the exponential term.

4

Express

2^{4x}

in terms of

2^{2x}

and evaluate.

2^{4x} = (2^{2x})^2 = 4^2 = 16

To find the final requested value using the exponent rule

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

.

Key Concept

Factoring exponential equations and applying power of a power exponent rules

Question 236Question →

A projectile is launched from a platform. Its height $h(t)$ , in meters, $t$ seconds after launch is modeled by the function $h(t) = -5t^2 + bt + 12$ , where $b$ is a positive constant. If the projectile reaches a maximum height of $32$ meters, what is the value of $b$ ?

Show answer & explanation

Answer: 20

Answer

The correct answer is 20.

To find the maximum height of the projectile, we locate the vertex of the quadratic function

h(t) = -5t^2 + bt + 12

. The

t

-coordinate of the vertex is given by

t = -\frac{B}{2A} = -\frac{b}{2(-5)} = \frac{b}{10}

. Substituting

t = \frac{b}{10}

into

h(t)

and setting the height to

32

meters gives

-5\left(\frac{b}{10}\right)^2 + b\left(\frac{b}{10}\right) + 12 = 32

. Simplifying this equation results in

-\frac{b^2}{20} + \frac{b^2}{10} = 20

, which reduces to

\frac{b^2}{20} = 20

. Solving for

b

gives

b^2 = 400

. Because

b

must be a positive constant,

b = 20

.

Step-by-Step Solution

1

Identify the time

t

at which the maximum height occurs using the vertex formula

t = -\frac{B}{2A}

for the quadratic function

h(t) = At^2 + Bt + C

.

t = \frac{b}{10}

The vertex of a downward-opening parabola represents its maximum value. For

h(t) = -5t^2 + bt + 12

, the coefficients are

A = -5

and

B = b

.

2

Substitute the time

t = \frac{b}{10}

back into the height function

h(t)

and set the expression equal to the maximum height of

32

meters.

-5\left(\frac{b}{10}\right)^2 + b\left(\frac{b}{10}\right) + 12 = 32

At the maximum height, the height of the projectile is

32

meters, which corresponds to the value of the function at the vertex time.

3

Simplify the equation to solve for

b^2

.

\frac{b^2}{20} = 20

Squaring the fraction yields

-5\left(\frac{b^2}{100}\right) + \frac{b^2}{10} + 12 = 32

. Simplifying the coefficients leads to

-\frac{b^2}{20} + \frac{b^2}{10} = 20

, which simplifies to

\frac{b^2}{20} = 20

.

4

Solve the equation for

b

.

b = 20

Multiplying both sides by

20

gives

b^2 = 400

. Taking the square root of both sides gives

b = \pm 20

. Since

b

is specified to be a positive constant,

b = 20

.

Key Concept

Finding the vertex of a quadratic function to determine maximum values in a real-world context.

Estimated Time:2m 0s

Question 237Question →

A point $(x, y)$ in the coordinate plane satisfies the system of inequalities below.

y \geq 2x - 4

y \leq -x + 5

x \geq 0

y \geq 0

What is the maximum possible value of $x$ for this point?

Show answer & explanation

Answer: 3

Answer

The maximum possible value of

x

that satisfies the system of inequalities is 3.

To find the maximum possible value of

x

for a point satisfying the system of inequalities, we identify the vertices of the region. The upper-right boundary of the region is formed by the intersection of the lines

y = 2x - 4

and

y = -x + 5

. Solving

2x - 4 = -x + 5

gives

3x = 9 \implies x = 3

. At this point,

y = -3 + 5 = 2

, which satisfies the constraints

x \geq 0

and

y \geq 0

. The other boundary vertices of the region are

(0, 0)

,

(2, 0)

, and

(0, 5)

. Comparing the x-coordinates of these vertices (

0

,

2

, and

3

), we see that the maximum possible value of

x

is

3

.

Step-by-Step Solution

1

Find the intersection point of the boundary lines

y = 2x - 4

and

y = -x + 5

.

x = 3

Setting the two boundary line equations equal to each other (

2x - 4 = -x + 5

) allows us to find the x-coordinate where the boundaries cross.

2

Substitute

x = 3

back into one of the boundary equations to find the y-coordinate.

y = 2

This yields the intersection point

(3, 2)

which lies on both boundary lines.

3

Verify that the intersection point

(3, 2)

satisfies the other constraints:

x \geq 0

and

y \geq 0

.

3 \geq 0

and

2 \geq 0

(both true)

The point must lie within the first quadrant to be a valid solution.

4

Determine the remaining boundary vertices of the solution region in the first quadrant.

Vertices are

(0, 0)

,

(0, 5)

,

(2, 0)

, and

(3, 2)

.

Comparing all vertices will confirm if

(3, 2)

indeed provides the maximum value of

x

.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 238Question →

\begin{aligned} x - y &= 3 \\ y &= x^2 - 7x + 12 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $y > 0$ , what is the value of $xy$ ?

Show answer & explanation

Answer: 10

Answer

10

By substituting the linear equation

y = x - 3

into the quadratic equation, we obtain a single quadratic equation

x^2 - 8x + 15 = 0

. Factoring this equation yields the solutions

x = 3

and

x = 5

. The corresponding

y

-coordinates are

y = 0

and

y = 2

, respectively. Since the problem specifies that

y > 0

, we choose the solution

(5, 2)

. The product of the coordinates is

5 \cdot 2 = 10

.

Step-by-Step Solution

1

Express

y

in terms of

x

from the linear equation.

y = x - 3

This allows for substitution into the second equation.

2

Substitute

y = x - 3

into the quadratic equation.

x - 3 = x^2 - 7x + 12

To create a single quadratic equation in terms of

x

.

3

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 8x + 15 = 0

Subtracting

x

and adding

3

to both sides enables factoring.

4

Factor the quadratic expression.

(x - 3)(x - 5) = 0

To find the roots of the equation, which are

x = 3

and

x = 5

.

5

Find the corresponding

y

-values for each

x

-value.

For

x = 3

,

y = 3 - 3 = 0

. For

x = 5

,

y = 5 - 3 = 2

.

To obtain the complete coordinate pairs of the intersection points.

6

Apply the constraint

y > 0

to select the correct solution pair.

The solution

(5, 2)

is selected since its

y

-coordinate is greater than

0

.

The other solution,

(3, 0)

, has

y = 0

, which does not satisfy the constraint

y > 0

.

7

Calculate the value of

xy

for the chosen solution.

xy = 5 \cdot 2 = 10

To find the final requested value.

Key Concept

Solving a nonlinear system of equations containing a linear equation and a quadratic equation by substitution.

Estimated Time:1m 30s

Question 239Question →

If $4^{x - 1} = 8$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2.5

Answer

2.5 (or 5/2)

The correct answer is 2.5 (or 5/2). By expressing 4 as

2^2

and 8 as

2^3

, the equation is rewritten as

(2^2)^{x-1} = 2^3

. Applying the exponent power rule simplifies this to

2^{2x-2} = 2^3

. Equating the exponents yields the linear equation

2x-2 = 3

. Solving for

x

gives

2x = 5

, which results in

x = 2.5

or

5/2

.

Step-by-Step Solution

1

Rewrite both sides of the equation with a common base of 2.

(2^2)^{x-1} = 2^3

Expressing both bases as powers of 2 allows us to equate the exponents later.

2

Apply the power of a power exponent rule

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

to the left side.

2^{2x-2} = 2^3

Multiplying the exponent 2 by the exponent

(x-1)

simplifies the expression to

2(x-1) = 2x-2

.

3

Equate the exponents and solve the linear equation for

x

.

x = 2.5

Since the bases are identical, their exponents must be equal, giving

2x-2 = 3

, which simplifies to

2x = 5

.

Key Concept

Solving exponential equations by finding a common base.

Question 240Question →

The function $f$ is defined by $f(x) = x^2 - 10x + 29$ . For what value of $x$ does $f(x)$ reach its minimum value?

Show answer & explanation

Answer: 5

Answer

The function reaches its minimum value at

x = 5

.

A quadratic function defined by

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

where

a > 0

reaches its minimum value at its vertex. The

x

-coordinate of the vertex is calculated using the formula

x = -\frac{b}{2a}

. For

f(x) = x^2 - 10x + 29

, the coefficients are

a = 1

and

b = -10

. Substituting these values into the formula gives

x = -\frac{-10}{2(1)} = 5

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic function in standard form

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

.

For

f(x) = x^2 - 10x + 29

, the coefficients are

a = 1

and

b = -10

.

To use the vertex formula

x = -\frac{b}{2a}

.

2

Apply the vertex formula to calculate the

x

-coordinate where the function reaches its minimum.

x = 5

.

Since the coefficient of

x^2

is positive (

a = 1

), the parabola opens upward, and its minimum value occurs at the vertex.

Key Concept

Finding the x-coordinate of the vertex of a quadratic function

Answer

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