Question 1

Choose the one alternative that best completes the statement or answers the question.
-The concentration of a drug in the bloodstream, measured in milligrams per liter, can be modeled by the function, $C ( t ) = \frac { 12 t + 4 } { 3 t ^ { 2 } + 2 }$, where $t$ is the number of minutes after injection of the drug. When will the drug be at its highest concentration? Approximate your answer rounded to two decimal places.

Accepted Answer

The answer of Choose the one alternative that best completes...

Question 2

Write the word or phrase that best completes each statement or answers the question.
-a A rare species of insect was discovered in the rain forest of Costa Rica. Environmentalists transplant the insect in protected area. The population of the insect $t$ months after being transplanted is
$$\mathrm { P } ( \mathrm { t } ) = \frac { 45 ( 1 + 0.6 \mathrm { t } ) } { ( 3 + 0.02 \mathrm { t } ) }$$
a) What was the population when $t = 0$ ?
b) What will the population be after 10 years?
c) What is the largest value the population could reach?

Accepted Answer

The answer of Write the word or phrase that best...

Question 3

Write the word or phrase that best completes each statement or answers the question.
-Which of the following functions could have this graph?

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Choose the one alternative that best completes the statement or answers the question.
-Economists use what is called a Leffer curve to predict the government revenue for tax rates from $0 \%$ to $100 \%$. Economists agree that the end points of the curve generate 0 revenue, but disagree on the tax rate that produces $t$ maximum revenue. Suppose an economist produces this rational function
$R ( x ) = \frac { 10 x ( 100 - x ) } { 25 + x }$, where $R$ is revenue in millions at a tax rate of $x$ percent. Use a graphing calculator to graph the function. What tax rate produces the maximum revenue? What is the maximum revenue?

Accepted Answer

The answer of Choose the one alternative that best completes...

Question 5

Choose the one alternative that best completes the statement or answers the question.
-A can in the shape of a right circular cylinder is required to have a volume of 700 cubic centimeters. The top and bottom are made up of a material that costs $8 \%$ per square centimeter, while the sides are made of material that costs 5 e per square centimeter. Which function below describes the total cost of the material as a function of the radius $\mathrm { r }$ of the cylinder?

Accepted Answer

The answer of Choose the one alternative that best completes...

Question 6

Solve the inequality.
-$$( x - 2 ) ( x - 3 ) > 0$$

Accepted Answer

The inequality $(x - 2)(x - 3) > 0$ holds true for values of $x$ that make the product positive. This occurs when $x$ is less than 2 or greater than 3, hence the correct intervals are $(-\infty, 2)$ and $(3, \infty)$.

Question 7

Solve the inequality.
-$$x ^ { 2 } + 7 x \geq 0$$

Accepted Answer

The inequality can be factored as $x(x + 7) \geq 0$, which is true when $x \leq -7$ or $x \geq 0$, corresponding to intervals $(- \infty, -7]$ and $[0, \infty)$.

Question 8

Choose the one alternative that best completes the statement or answers the question.
-A closed box with a square base has to have a volume of 16,000 cubic inches. Find a function for the surface area of the box.

Accepted Answer

The surface area $S$ of a box with a square base and volume $V = 16,000$ cubic inches can be found using the formula for the volume $V = x^2h$, where $x$ is the side length of the square base and $h$ is the height. Solving for $h$, we get $h = \frac{V}{x^2} = \frac{16,000}{x^2}$. The surface area of the box is the sum of the areas of the four sides and the two bases: $S = 2xh + 2xh + 2x^2 = 4xh + 2x^2$. Substituting $h$ gives $S = 4x(\frac{16,000}{x^2}) + 2x^2 = \frac{64,000}{x} + 2x^2$, which matches choice B.

Question 9

Solve the inequality.
-$$x ^ { 2 } - 64 > 0$$

Accepted Answer

This inequality can be solved by factoring the left side as a difference of squares: $x^2 - 64 = (x + 8)(x - 8) > 0$. The critical points are $x = -8$ and $x = 8$. Testing intervals around these points shows that the inequality is satisfied for $x < -8$ and $x > 8$, leading to the solution $(-\infty, -8)$ or $(8, \infty)$.

Question 10

Solve the problem.
-Determine which rational function $\mathrm { R } ( \mathrm { x } )$ has a graph that crosses the $x$-axis at $- 1$, touches the $x$-axis at $- 4$, has vertical asymptotes at $x = - 2$ and $x = 3$, and has one horizontal asymptote at $y = - 2$.

Accepted Answer

The correct function must have factors that correspond to its zeros and asymptotes. For $x$-axis crossings, $x+1$ and $(x+4)^2$ are needed, with the square on $(x+4)$ indicating the touch point at $-4$. Vertical asymptotes at $x=-2$ and $x=3$ require these factors in the denominator, and the leading coefficient and highest power of $x$ in the numerator and denominator determine the horizontal asymptote at $y=-2$, making option A the correct choice.

Question 11

Write the word or phrase that best completes each statement or answers the question.
-A box has a base whose length is twice its width. The volume of the box is 7000 cubic inches.
a) Find a function for the surface area of the box.
b) What are the dimensions of the box that minimizes surface area?

Accepted Answer

The answer of Write the word or phrase that best...

Question 12

Choose the one alternative that best completes the statement or answers the question.
-Economists use what is called a Leffer curve to predict the government revenue for tax rates from $0 \%$ to $100 \%$. Economists agree that the end points of the curve generate 0 revenue, but disagree on the tax rate that produces $t$ maximum revenue. Suppose an economist produces this rational function
$R ( x ) = \frac { 10 x ( 100 - x ) } { 15 + x }$, where $R$ is revenue in millions at a tax rate of x percent. Use a graphing calculator to graph the function. What tax rate produces the maximum revenue? What is the maximum revenue?

Accepted Answer

The answer of Choose the one alternative that best completes...

Question 13

Solve the inequality.
-$$x ^ { 2 } - 5 x \leq 0$$

Accepted Answer

The answer of Solve the inequality.
-\[x ^ { 2 }...

Question 14

Solve the inequality.
-$$x ^ { 2 } + 6 x \leq 0$$

Accepted Answer

To solve the inequality $x^2 + 6x \leq 0$, factor it to $x(x + 6) \leq 0$. The roots are $x = 0$ and $x = -6$. The inequality holds between these roots, including the endpoints, hence the solution is $[ - 6,0 ]$.

Question 15

Choose the one alternative that best completes the statement or answers the question.
-Economists use what is called a Leffer curve to predict the government revenue for tax rates from $0 \%$ to $100 \%$. Economists agree that the end points of the curve generate 0 revenue, but disagree on the tax rate that produces t maximum revenue. Suppose an economist produces this rational function
$R ( x ) = \frac { 10 x ( 100 - x ) } { 75 + x }$, where $R$ is revenue in millions at a tax rate of $x$ percent. Use a graphing calculator to graph the function. What tax rate produces the maximum revenue? What is the maximum revenue?

Accepted Answer

To find the tax rate that produces the maximum revenue and the maximum revenue itself, one would typically use calculus or a graphing calculator to analyze the function $R(x) = \frac{10x(100-x)}{75+x}$. However, without performing the calculations directly here, the correct answer is based on the premise of the question. The maximum revenue and corresponding tax rate can be determined by graphing the function and finding the vertex of the curve or by using optimization techniques. Among the given options, $39.6\%$ tax rate producing \$209 million in revenue is identified as the correct choice based on the context provided.

Question 16

Choose the one alternative that best completes the statement or answers the question.
-Economists use what is called a Leffer curve to predict the government revenue for tax rates from $0 \%$ to $100 \% .$ Economists agree that the end points of the curve generate 0 revenue, but disagree on the tax rate that produces t maximum revenue. Suppose an economist produces this rational function
$R ( x ) = \frac { 10 x ( 100 - x ) } { 50 + x }$, where $R$ is revenue in millions at a tax rate of $x$ percent. Use a graphing calculator to graph the function. What tax rate produces the maximum revenue? What is the maximum revenue?

Accepted Answer

To find the maximum revenue and the tax rate that produces it, you would typically use calculus or a graphing calculator to analyze the function $R(x) = \frac{10x(100-x)}{50+x}$. By graphing this function and finding its maximum point, you can determine that the tax rate which produces the maximum revenue is approximately $36.6\%$ and the maximum revenue itself is about \$268 million. This aligns with option C.

Question 17

Solve the inequality.
-$$\mathrm { x } ^ { 2 } - 36 \leq 0$$

Accepted Answer

The inequality can be rewritten as $(x-6)(x+6) \leq 0$, which holds for $x$ in the interval $[-6, 6]$.

Question 18

Solve the inequality.
-$$( x + 2 ) ( x - 7 ) \leq 0$$

Accepted Answer

The inequality $(x + 2)(x - 7) \leq 0$ holds true when one factor is non-positive and the other is non-negative, which occurs between the roots $x = -2$ and $x = 7$, inclusive.

Question 19

Write the word or phrase that best completes each statement or answers the question.
-The concentration C of a certain drug in a patient's bloodstream is given by $$\frac { 30 t } { t ^ { 2 } + 49 }$$ a) Find the horizontal asymptote of C(t).
b) Using a graphing utility, determine the time at which the concentration is highest.

Accepted Answer

The answer of Write the word or phrase that best...

Question 20

Solve the inequality.
-$$x ^ { 2 } - 8 x \geq 0$$

Accepted Answer

The answer of Solve the inequality.
-\[x ^ { 2 }...

Quiz 3: Polynomial and Rational Functions

Solve the inequality. - $( x - 2 ) ( x - 3 ) > 0$

Solve the inequality. - $x ^ { 2 } + 7 x \geq 0$

Choose the one alternative that best completes the statement or answers the question. -A closed box with a square base has to have a volume of 16,000 cubic inches. Find a function for the surface area of the box.

Solve the inequality. - $x ^ { 2 } - 64 > 0$

Solve the problem. -Determine which rational function $\mathrm { R } ( \mathrm { x } )$ has a graph that crosses the $x$ -axis at $- 1$ , touches the $x$ -axis at $- 4$ , has vertical asymptotes at $x = - 2$ and $x = 3$ , and has one horizontal asymptote at $y = - 2$ .

Write the word or phrase that best completes each statement or answers the question. -A box has a base whose length is twice its width. The volume of the box is 7000 cubic inches. a) Find a function for the surface area of the box. b) What are the dimensions of the box that minimizes surface area?

Solve the inequality. - $x ^ { 2 } - 5 x \leq 0$

Solve the inequality. - $x ^ { 2 } + 6 x \leq 0$

Solve the inequality. - $\mathrm { x } ^ { 2 } - 36 \leq 0$

Solve the inequality. - $( x + 2 ) ( x - 7 ) \leq 0$

Solve the inequality. - $x ^ { 2 } - 8 x \geq 0$

Quiz 3: Polynomial and Rational Functions

Write the word or phrase that best completes each statement or answers the question. -Which of the following functions could have this graph?

Solve the inequality. - (x−2) (x−3) >0( x - 2 ) ( x - 3 ) > 0(x−2) (x−3) >0

Solve the inequality. - x2+7x≥0x ^ { 2 } + 7 x \geq 0x2+7x≥0

Choose the one alternative that best completes the statement or answers the question. -A closed box with a square base has to have a volume of 16,000 cubic inches. Find a function for the surface area of the box.

Solve the inequality. - x2−64>0x ^ { 2 } - 64 > 0x2−64>0

Write the word or phrase that best completes each statement or answers the question. -A box has a base whose length is twice its width. The volume of the box is 7000 cubic inches. a) Find a function for the surface area of the box. b) What are the dimensions of the box that minimizes surface area?

Solve the inequality. - x2−5x≤0x ^ { 2 } - 5 x \leq 0x2−5x≤0

Solve the inequality. - x2+6x≤0x ^ { 2 } + 6 x \leq 0x2+6x≤0

Solve the inequality. - x2−36≤0\mathrm { x } ^ { 2 } - 36 \leq 0x2−36≤0

Solve the inequality. - (x+2) (x−7) ≤0( x + 2 ) ( x - 7 ) \leq 0(x+2) (x−7) ≤0

Solve the inequality. - x2−8x≥0x ^ { 2 } - 8 x \geq 0x2−8x≥0

Solve the inequality. - $( x - 2 ) ( x - 3 ) > 0$

Solve the inequality. - $x ^ { 2 } + 7 x \geq 0$

Solve the inequality. - $x ^ { 2 } - 64 > 0$

Solve the inequality. - $x ^ { 2 } - 5 x \leq 0$

Solve the inequality. - $x ^ { 2 } + 6 x \leq 0$

Solve the inequality. - $\mathrm { x } ^ { 2 } - 36 \leq 0$

Solve the inequality. - $( x + 2 ) ( x - 7 ) \leq 0$

Solve the inequality. - $x ^ { 2 } - 8 x \geq 0$