Chapter 9

Air is being pumped into a spherical balloon at the rate of 100 \(\mathrm{cm}^{3} / \mathrm{sec} .\) How fast is the diameter increasing when the radius is \(5 \mathrm{~cm} ?\)

Two cars leave an intersection at the same time. The first car is going due east at the rate of \(40 \mathrm{mph}\) and the second is going due south at the rate of \(30 \mathrm{mph}\). How fast is the distance between the two cars increasing when the first car is 120 miles from the intersection?

Find a number in the interval (0,2) such that the sum of the number and its reciprocal is the absolute minimum.

An open box is to be made using a piece of cardboard \(8 \mathrm{~cm}\) by \(15 \mathrm{~cm}\) by cutting a square from each corner and folding the sides up. Find the length of a side of the square being cut so that the box will have a maximum volume.

What is the shortest distance between the \(\left(2,-\frac{1}{2}\right)\) and the parabola \(y=-x^{2} ?\)

A man with 200 meters of fence plans to enclose a rectangular piece of land using a river on one side and a fence on the other three sides. Find the maximum area that the man can obtain.

A plane lifts off from a runway at an angle of \(20^{\circ}\). If the speed of the plane is \(300 \mathrm{mph}\), how fast is the plane gaining altitude?

Given the cost function \(\mathrm{C}(x)=2500+0.02 x+0.004 x^{2},\) find the product level such that the average cost per unit is a minimum.

Find the maximum area of a rectangle inscribed in an ellipse whose equation is \(4 x^{2}+25 y^{2}=100\).

A right triangle is in the first quadrant with a vertex at the origin and the other two vertices on the \(x-\) and \(y\) -axes. If the hypotenuse passes through the point \((0.5,4),\) find the vertices of the triangle so that the length of the hypotenuse is the shortest possible length.

If \(y=\sin ^{2}(\cos (6 x-1)),\) find \(\frac{d y}{d x}\).

Evaluate \(\lim _{x \rightarrow \infty} \frac{100 / x}{-4+x+x^{2}} .\)

(Calculator) At what value(s) of \(x\) does the tangent to the curve \(x^{2}+\) \(y^{2}=36\) have a slope of \(-1 ?\)

(Calculator) Find the shortest distance between the point (1,0) and the curve \(y=x^{3}\).