91Ó°ÊÓ

An open box of maximum volume is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum volume. $$ \begin{array}{|c|c|c|} \hline \text { Height } & \begin{array}{c} \text { Length and } \\ \text { Width } \end{array} & \text { Volume } \\ \hline 1 & 24-2(1) & 1[24-2(1)]^{2}=484 \\ \hline 2 & 24-2(2) & 2[24-2(2)]^{2}=800 \\ \hline \end{array} $$ (b) Write the volume \(V\) as a function of \(x\). (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

Determine the open intervals on which the graph is concave upward or concave downward. \(y=-x^{3}+3 x^{2}-2\)

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=x^{2}-5 x+4,[1,4] $$

Area The measurement of a side of a square is found to be 15 centimeters, with a possible error of 0.05 centimeter. (a) Approximate the percent error in computing the area of the square. (b) Estimate the maximum allowable percent error in measuring the side if the error in computing the area cannot exceed \(2.5 \%\)

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=x-2 \ln x,[1,3] $$

Circumference \(\quad\) The measurement of the circumference of a circle is found to be 60 centimeters, with a possible error of 1.2 centimeters. (a) Approximate the percent error in computing the area of the circle. (b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed \(3 \%\).

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow \infty}\left(4+\frac{3}{x}\right) $$

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

{Volume and Surface Area } The radius of a sphere is measured to be 6 inches, with a possible error of 0.02 inch. Use differentials to approximate the maximum possible error in calculating (a) the volume of the sphere, (b) the surface area of the sphere, and (c) the relative errors in parts (a) and (b).

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 337.5 square centimeters.