91Ó°ÊÓ

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100\(\pi\) feet?

Find the volume of the solid that results when the region bounded by \(y=\sqrt{9-x^{2}}\) and the \(x\) -axis is revolved around the \(x\) -axis.

Find the average value of \(f(x)=\sqrt{x}\) on the interval \([0,16]\)

Find the velocity and acceleration of a particle whose position function is \(x(t)=\sin (2 t)+\cos (t)\)

A spherical balloon is inflating at a rate of \(27 \pi \mathrm{in}^{3} / \mathrm{sec} .\) How fast is the radius of the balloon increasing when the radius is 3 inches?

Now find the derivative of each of the following functions. \(f(x)=\ln (\cot x-\csc x)\)

Is the function \(f(x)=\sec x\) continuous everywhere?

A 384 -square-meter plot of land is to be enclosed by a fence and divided into two equal parts by another fence parallel to one pair of sides. What dimensions of the outer rectangle will minimize the amount of fence used?

Cars A and B leave a town at the same time. Car A heads due south at a rate of 80 km/hr and car B heads due west at a rate of 60 km/hr. How fast is the distance between the cars increasing after three hours?

Water is draining at the rate of 48\(\pi \mathrm{f}^{3} / \mathrm{second}\) from the vertex at the bottom of a conical tank whose diameter at its base is 40 feet and whose height is 60 feet. (a) Find an expression for the volume of water in the tank, in terms of its radius, at the surface of the water. (b) At what rate is the radius of the water in the tank shrinking when the radius is 16 feet? (c) How fast is the height of the water in the tank dropping at the instant that the radius is 16 feet?