Exam 4: Rational, Power, and Root Functions

(a) Solve the following rational equation analytically: $\frac{1}{x+2}-\frac{6}{x-2}=\frac{6}{x^{2}-4}$ . (b) Use the results of part (a) and a graph to find the solution set of $\frac{1}{x+2}-\frac{6}{x-2} \leq \frac{6}{x^{2}-4}$ .

The parking attendants working at the exit of a parking ramp can process at most 15 cars per minute. If cars arrive randomly at an average rate of $x$ vehicles per minute, then the average wait $W$ in minutes for a car to exit the ramp is approximated by $W(x)=\frac{1}{15-x}$ , where $0 \leq x<15$ . (a) Evaluate $W(10), W(14)$ , and $W(14.9)$ . Interpret the results. (b) Graph $W$ using the window $[0,15]$ by $[-.5,1]$ . Identify the vertical asymptote. What happens to $W$ as $x$ approaches 15 ? (c) Find $x$ when the wait is 2 minutes.

A manufacturer needs to construct a box with a lid for a special product. The only stipulations are that the volume of the box should be 2515 cubic centimeters and that the box should have a square base. The cost for producing such a box has been determined to be represented by the function $C(x)=\frac{100+0.02 x^{3}}{x}$ , where $C(x)$ is the cost of the box in dollars and $x$ is the length of a side of the base in centimeters. Use the graph of $C$ to determine the side length $x$ that will minimize the cost of the box, and determine what this cost will be. (Hint: Use the window $[0,50]$ by $[0,50]$ .)

(a) Sketch the graph of $f(x)=-\frac{1}{x^{2}}+2$ (b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x^{2}}$ . (c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

The length and cross-sectional area of a wire determine the resistance that the wire gives to the flow of electricity. The resistance of a wire varies directly as the length of the wire and inversely as the cross-sectional area. A wire with a length of $2400 \mathrm{~cm}$ and a cross-sectional area of $6.29 \times 10^{-3} \mathrm{~cm}^{2}$ has a resistance of $.656 \mathrm{ohm}$ . If another wire has a length of $2000 \mathrm{~cm}$ and a cross-sectional area of $7.11 \times 10^{-3} \mathrm{~cm}^{2}$ , determine the resistance of the wire.

Two buildings are situated on level ground, 65 feet apart, as shown in the figure. The building on the left is 30 feet high and the other building is 45 feet high. An expensive decorative banner is to be strung from the top edge of one building to the top edge of the other, pulled tightly so that it touches the ground at point $P$ somewhere between the two buildings. Let $x$ represent the distance from $P$ to the base of the building on the left. Find the value of $x$ that will minimize the length of the banner. How long is this banner?

Consider the rational function defined by $f(x)=\frac{x^{2}-25}{x+5}$ . (a) For what value of $x$ does the graph exhibit a "hole"? (b) Graph the function and show the "hole" in the graph.

(a) Solve the equation $\sqrt{9-x}=x-3$ analytically. Support the solution(s) with a graph. (b) Use the graph to find the solution set of $\sqrt{9-x} \leq x-3$ . (c) Use the graph to find the solution set of $\sqrt{9-x}>x-3$ .