91Ó°ÊÓ

Minimizing perimeter What is the smallest perimeter possible for a rectangle whose area is 16 in \(^{2},\) and what are its dimensions?

A rectangle has its base on the \(x\) -axis and its upper two vertices on the parabola \(y=12-x^{2} .\) What is the largest area the rectangle can have, and what are its dimensions?

Use Newton's method to find the negative fourth root of 2 by solving the equation \(x^{4}-2=0 .\) Start with \(x_{0}=-1\) and find \(x_{2}\).

The shortest fence \(A\) 216 \(\mathrm{m}^{2}\) rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed?

Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3 GRAPH CANT COPY

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\sec ^{2} x\) b. \(\frac{2}{3} \sec ^{2} \frac{x}{3}\) c. \(-\sec ^{2} \frac{3 x}{2}\)

a. The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume? GRAPH CANT COPY b. Graph the volume of a 108 -in. box (length plus girth equals 108 in.) as a function of its length and compare what you see with your answer in part (a). (Continuation of Exercise \(20 .\) ) a. Suppose that instead of having a box with square ends you have a box with square sides so that its dimensions are \(h\) by \(h\) by \(w\) and the girth is \(2 h+2 w .\) What dimensions will give the box its largest volume now?b. Graph the volume as a function of \(h\) and compare what you see with your answer in part (a). GRAPH CANT COPY

A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is twice as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction.

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(x-\left(\frac{1}{2}\right)^{x}\) b. \(x^{2}+2^{x}\) c. \(\pi^{x}-x^{-1}\)

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$f(x)=4-x^{3},-2 \leq x \leq 1$$