91Ó°ÊÓ

The postal service places a limit of 84 in. on the combined length and girth of (distance around) a package to be sent parcel post. What dimensions of a rectangular box with square cross-section will contain the largest volume that can be mailed? (Hint: There are two different girths.)

A rectangular play area of \(48 \mathrm{yd}^{2}\) is to be fenced off in a person's yard. The next-door neighbor agrees to pay half the cost of the fence on the side of the play area that lies along the property line. What dimensions will minimize the cost of the fence?

Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. $$ f(x)=\frac{x^{3}}{x^{2}-1} $$

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$f(x)=x(25-x)$$

Suppose a rope surrounds the earth at the equator. The rope is lengthened by \(10 \mathrm{ft}\). By about how much is the rope raised above the earth?

Cities and companies find that the cost of pollution control increases along with the percentage of pollutants being removed. Suppose the \(\operatorname{cost} C,\) in dollars, of removing \(p \%\) of the pollutants from a chemical spill is given by \(C(p)=\frac{48,000}{100-p}\). a) Find \(C(0), C(20), C(80),\) and \(C(90)\). b) Find the domain of \(C\). c) Draw a graph of \(C\). d) Can the company or city afford to remove \(100 \%\) of the pollutants due to this spill? Explain.

After an injection, the amount of a medication \(A\), in cubic centimeters (cc), in the bloodstream decreases with time \(t,\) in hours. Suppose that under certain conditions \(A\) is given by \(A(t)=\frac{A_{0}}{t^{2}+1},\) where \(A_{0}\) is the initial amount of the medication. Assume that an initial amount of \(100 \mathrm{cc}\) is injected. a) Find \(A(0), A(1), A(2), A(7),\) and \(A(10)\). b) Find the maximum amount of medication in the bloodstream over the interval \([0, \infty)\). c) Graph the function. d) According to this function, does the medication ever completely leave the bloodstream? Explain your answer.

Use calculus to prove that the relative minimum or maximum for any function \(f\) for which $$ f(x)=a x^{2}+b x+c, \quad a \neq 0 $$ occurs at \(x=-b /(2 a)\).

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$f(x)=x^{2}+\frac{250}{x} ; \quad(0, \infty)$$

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$f(x)=\sqrt{x} ; \quad[0,4]$$