Question 1

Given the point $$P(x, y) on $$the graph of $$y = x^2$ - 3$$, express the distance $$d$$ from $$P to $$the origin $$(0, 0) as a $$function of $$x.$$

Accepted Answer

$$d(x) = \sqrt{x^2$ + (x^2$ - 3)^2$}$$

Question 2

For the function $$d(x) = \sqrt{x^2$ + (x^2$ - 3)^2$}$$, what is the value of $$d$$ when $$x = 0$$?

Accepted Answer

$$d = 3$$

Question 3

For the function $$d(x) = \sqrt{x^2$ + (x^2$ - 3)^2$}$$, what is the value of $$d$$ when $$x = 2$$?

Accepted Answer

$$d = \sqrt{2^2$ + (2^2$ - 3)^2$} = \sqrt{4 + 1^2$} = \sqrt{5}$$

Question 4

A rectangle has one corner at the origin, one on the positive $$x-$$axis at $$(x, 0)$$, one on the positive $$y-$$axis at $$(0, y)$$, and one in quadrant I on the graph $$y = 16 - x^2$. Express the area A$ of the rectangle as a function of x$.

Accepted Answer

$$A(x) = x(16 - x^2$)$$

Question 5

What is the domain of the area function $$A(x) = x(16 - x^2$)$$ for the rectangle described above?

Accepted Answer

$$0 \u003c x \u003c 4$$

Question 6

Suppose you are designing a rectangular garden with one side along a river (no fence needed there). If you have 100 meters of fencing for the other three sides, express the area $$A as a $$function of the width $$x$$ perpendicular to the river.

Accepted Answer

$$A(x) = x(100 - 2x)$$

Question 7

A company’s profit $$P (in $$thousands of dollars) is modeled by $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units produced (in thousands). What is the maximum profit, and for what value of $$x$$ does it occur?

Accepted Answer

Maximum profit is $$-2\left(10\$$right)^2$$ + 40\left(10\right) - 100 = 100 at$$ $$x = 10$$

Question 8

Given the function $$f(x) = \sqrt{x^2$ + (x^2$ - 3)^2$}$$, which of the following best describes the range of $$f(x)$$ for all real $$x$$?

Accepted Answer

$$[3, \infty)$$

Question 9

If the area of a rectangle is given by $$A(x) = x(16 - x^2$)$$, for what value of $$x is $$the area maximized?

Accepted Answer

$$x = \sqrt{16/3}$$

Question 10

A ball is thrown upward from the ground with an initial velocity of 32 ft/s. Its height after $$t$$ seconds is given by $$h(t) = 32t - 16t^2$. What is the maximum height reached by the ball?

Accepted Answer

$$h = 16 ft$$

Question 11

A company wants to build a box with a square base and an open top. If the volume must be 500 cubic units, express the surface area $$S as a $$function of the base side length $$x.$$

Accepted Answer

$$S(x) = x^2$ + \frac{2000}{x}$$

Question 12

Suppose the cost $$C (in $$dollars) to produce $$x$$ units of a product is given by $$C(x) = 5x + 200. $$What is the cost to produce 50 units?

Accepted Answer

$$C = 450$$

Question 13

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. What is the population after 20 hours?

Accepted Answer

$$P = 100e$$^{1}$$ \approx 271.83$$

Question 14

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.2t}$$, where t$ is in hours. What is the concentration after 5 hours?

Accepted Answer

$$C = 50e$$^{-1}$$ \approx 18.4$$

Mathematical Models: Building and Analyzing Functions

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