91Ó°ÊÓ

Graph each function using the Guidelines for Graphing Rational Functions, which is simply modified to include nonlinear asymptotes. Clearly label all intercepts and asymptotes and any additional points used to sketch the graph. $$Y_{1}=\frac{x^{3}-3 x+2}{x^{2}-9}$$

The cost to produce bottled spring water is given by \(C(x)=16 x-63,\) where \(x\) is the number of thousands of bottles. The total income (revenue) from the sale of these bottles is given by the function \(R(x)=-x^{2}+326 x-7463\). since profit \(=\) revenue \(-\) cost, the profit function must be \(P(x)=-x^{2}+310 x-7400\) (verify). how many bottles sold will produce the maximum profit? What is the maximum profit?

Tina and Imai have just purchased a purebred German Shepherd, and need to fence in their backyard so the dog can run. What is the maximum rectangular area they can enclose with \(200 \mathrm{ft}\) of fencing, if (a) they use fencing material along all four sides? What are the dimensions of the rectangle? (b) What is the maximum area if they use the house as one of the sides? What are the dimensions of this rectangle?

The surface area of a spherical cap is given by \(S=2 \pi r h,\) where \(r\) is the radius of the sphere and \(h\) is the perpendicular distance from the sphere's surface to the plane intersecting the sphere, forming the cap. The volume of the cap is \(V=\frac{1}{3} \pi h^{2}(3 r-h) .\) Similar to Exercise \(61,\) a formula can be found that will minimize the area of a cap that holds a specified volume. a. Solve the volume formula for the variable \(r\) b. Substitute the resulting expression for \(r\) into the surface area formula and simplify. The result is a formula for surface area given solely in terms of the volume \(V\) and the height \(h\). c. Assume the volume of the spherical cap is \(500 \mathrm{cm}^{3} .\) Use a graphing calculator to graph the resulting function on an appropriate window, and use the graph to find the height \(h\) that will result in a spherical cap with the smallest possible area, while still holding a volume of \(500 \mathrm{cm}^{3}\) d. Use this value of \(h\) and \(V=500 \mathrm{cm}^{3}\) to find the radius of the sphere.

Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. $$\frac{x^{3}+1}{x^{2}+1}>0$$

A large group of students is asked to memorize a list of 50 Italian words, a language that is unfamiliar to them. The group is then tested regularly to see how many of the words are retained over a period of time. The average number of words retained is modeled by the function \(W(t)=\frac{6 t+40}{t},\) where \(W(t)\) represents the number of words remembered after \(t\) days. a. Graph the function over the interval \(t \in[0,40] .\) How many days until only half the words are remembered? How many days until only one-fifth of the words are remembered? b. After 10 days, what is the average number of words retained? How many days until only 8 words can be recalled? c. What is the significance of the horizontal asymptote (what does it mean in this context)?

The volume of water in a rectangular, in-ground, swimming pool is given by \(V(x)=x^{3}+11 x^{2}+24 x,\) where \(v(x)\) is the volume in cubic feet when the water is \(x\) ft high. (a) Use the remainder theorem to find the volume when \(x=3 \mathrm{ft}\). (b) If the volume is \(100 \mathrm{ft}^{3}\) of water, what is the height \(x ?\) (c) If the maximum capacity of the pool is \(1000 \mathrm{ft}^{3},\) what is the maximum depth (to the nearest integer)?

Use Descartes' rule of signs to determine the possible combinations of real and complex zeroes for each polynomial. Then graph the function on the standard window of a graphing calculator and adjust it as needed until you're certain all real zeroes are in clear view. Use this screen and a list of the possible rational zeroes to factor the polynomial and find all zeroes (real and complex). $$H(x)=4 x^{3}+60 x^{2}+53 x-42$$