Chapter 4

A storage tank for propane gas is to be constructed in the shape of a right circular cylinder of altitude 10 feet with a hemisphere attached to each end. Determine the radius \(x\) so that the resulting volume is \(27 \pi \mathrm{ft}^{3}\). (See Example 8 of Section 3.4.)

Find the oblique asymptote, and sketch the graph of \(f\). $$ f(x)=\frac{8-x^{3}}{2 x^{2}} $$

A storage shelter is to be constructed in the shape of a cube with a triangular prism forming the roof (see the figure). The length \(x\) of a side of the cube is yet to be determined. (a) If the total height of the structure is 6 feet, show that its volume \(V\) is given by \(V=x^{3}+\frac{1}{2} x^{2}(6-x)\). (b) Determine \(x\) so that the volume is \(80 \mathrm{ft}^{3}\).

Find the oblique asymptote, and sketch the graph of \(f\). $$ f(x)=\frac{x^{3}+1}{x^{2}-9} $$

A canvas camping tent is to be constructed in the shape of a pyramid with a square base. An 8-foot pole will form the center support, as illustrated in the figure. Find the length \(x\) of a side of the base so that the total amount of canvas needed for the sides and bottom is \(384 \mathrm{ft}^{2}\).

Simplify \(f(x)\), and sketch the graph of \(f\). $$ f(x)=\frac{2 x^{2}+x-6}{x^{2}+3 x+2} $$

Simplify \(f(x)\), and sketch the graph of \(f\). $$ f(x)=\frac{x^{2}-x-6}{x^{2}-2} $$

Simplify \(f(x)\), and sketch the graph of \(f\). $$ f(x)=\frac{x-1}{1-x^{2}} $$

Simplify \(f(x)\), and sketch the graph of \(f\). $$ f(x)=\frac{x+2}{x^{2}-4} $$

Simplify \(f(x)\), and sketch the graph of \(f\). $$ f(x)=\frac{x^{2}+x-2}{x+2} $$

Simplify \(f(x)\), and sketch the graph of \(f\). $$ f(x)=\frac{x^{3}-2 x^{2}-4 x+8}{x-2} $$

Simplify \(f(x)\), and sketch the graph of \(f\). $$ f(x)=\frac{x^{2}+4 x+4}{x^{2}+3 x+2} $$

Simplify \(f(x)\), and sketch the graph of \(f\). $$ 4 f(x)=\frac{\left(x^{2}+x\right)(2 x-1)}{\left(x^{2}-3 x+2\right)(2 x-1)} $$

vertical asymptotes: \(x=-2, x=0\) horizontal asymptote: \(y=0\) \(x\)-intercept: \(2 ; f(3)=1\)

Drug dosage Young's rule is a formula that is used to modify adult drug dosage levels for young children. If a denotes the adult dosage (in milligrams) and if \(t\) is the age of the child (in years), then the child's dose \(y\) is given by the equation \(y=t a /(t+12)\). Sketch the graph of this equation for \(t>0\) and \(a=100\).

Salt concentration Salt water of concentration 0.1 pound of salt per gallon flows into a large tank that initially conains 50 gallons of pure water. (a) If the flow rate of salt water into the tank is \(5 \mathrm{gal} / \mathrm{min}\), find the volume \(V(t)\) of water and the amount \(A(t)\) of salt in the tank after \(t\) minutes.

Salmon propagation For a particular salmon population, the relationship between the number \(S\) of spawners and the number \(R\) of offspring that survive to maturity is given by the formula $$ R=\frac{4500 S}{S+500} $$ (a) Under what conditions is \(R>S\) ? (b) Find the number of spawners that would yield \(90 \%\) of the greatest possible number of offspring that survive to maturity. (c) Work part (b) with \(80 \%\) replacing \(90 \%\). (d) Compare the results for \(S\) and \(R\) (in terms of percentage increases) from parts (b) and (c).

Population density The population density \(D\) (in people/ \(\mathrm{mi}^{2}\) ) in a large city is related to the distance \(x\) (in miles) from the center of the city by $$ D=\frac{5000 x}{x^{2}+36} $$ (a) What happens to the density as the distance from the center of the city changes from 20 miles to 25 miles? (b) What eventually happens to the density? (c) In what areas of the city does the population density exceed 400 people/ \(\mathrm{mi}^{-2}\) ?

Grade point average (GPA) (a) A student has finished 48 credit hours with a GPA of 2.75. How many additional credit hours \(y\) at \(4.0\) will raise the student's GPA to some desired value \(x\) ? (Determine \(y\) as a function of \(x\).) (b) Create a table of values for \(x\) and \(y\), starting with \(x=2.8\) and using increments of \(0.2\). (c) Graph the function in part (a). (d) What is the vertical asymptote of the graph in part (c)? (e) Explain the practical significance of the value \(x=4\).