91Ó°ÊÓ

A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass for the herd, the pasture must contain 180,000 square meters. No fencing is required along the river. What dimensions will use the least amount of fencing?

Find all relative extrema of the function. Use the Second-Derivative Test when applicable. \(f(x)=\sqrt{x^{2}+1}\)

An open box is to be made from a two-foot by three-foot rectangular piece of material by cutting equal squares from the corners and turning up the sides. Find the volume of the largest box that can be made in this manner.

A right triangle is formed in the first quadrant by the \(x-\) and \(y\) -axes and a line through the point \((1,2)\) (see figure). (a) Write the length \(L\) of the hypotenuse as a function of \(x\) (b) Use a graphing utility to approximate \(x\) graphically such that the length of the hypotenuse is a minimum. (c) Find the vertices of the triangle such that its area is a minimum.

Find the speed \(v,\) in miles per hour, that will minimize costs on a 110 -mile delivery trip. The cost per hour for fuel is \(C\) dollars, and the driver is paid \(W\) dollars per hour. (Assume there are no costs other than wages and fuel.) $$ \begin{array}{l}{\text { Fuel cost: } C=\frac{v^{2}}{300}} \\ {\text { Driver: } W=\$ 12}\end{array} $$

Find the price elasticity of demand for the demand function at the indicated -value. Is the demand elastic, inelastic, or of unit elasticity at the indicated -value? Use a graphing utility to graph the revenue function, and identify the intervals of elasticity and inelasticity. $$ p=5-0.03 x \quad x=100 $$

Find the price elasticity of demand for the demand function at the indicated -value. Is the demand elastic, inelastic, or of unit elasticity at the indicated -value? Use a graphing utility to graph the revenue function, and identify the intervals of elasticity and inelasticity. $$ p=20-0.0002 x \quad x=30 $$

Maximum Volume A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package with maximum volume. Assume that the package's dimensions are \(x\) by \(x\) by \(y\).

The sum of the perimeters of a circle and a square is 16. Find the dimensions of the circle and square that produce a minimum total area.

An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter running track. Find the dimensions that will make the area of the rectangular region as large as possible.