91Ó°ÊÓ

An inverted conical container has a diameter of 42 inches and a depth of 15 inches. If water is flowing out of the vertex of the container at a rate of 35\(\pi\) in 3/sec , how fast is the depth of the water dropping when the height is 5 in?

Try these 22 problems to test your skill with limits. $$ \lim _{x \rightarrow \infty}\left(\frac{x^{4}-8}{10 x^{4}+25 x+1}\right)= $$

Use the differential formulas in this chapter to solve these problems. When a spherical ball bearing is heated, its radius increases by 0.01 \(\mathrm{mm}\) . Estimate the change in volume of the ball bearing when the radius is 5 \(\mathrm{mm}\) .

Find the closest point on the curve \(x^{2}+y^{2}=1\) to the point \((2,1)\).

A boat is being pulled toward a dock by a rope attached to its bow through a pulley on the dock 7 feet above the bow. If the rope is hauled in at a rate of 4 ft/sec, how fast is the boat approaching the dock when 25 ft of rope is out?

If the position function of a particle is \(x(t)=2 \sin ^{2} t+2 \cos ^{2} t, t>0,\) find the velocity and acceleration of the particle.

Use the differential formulas in this chapter to solve these problems. A cylindrical tank is constructed to have a diameter of 5 meters and a height of 20 meters. Find the error in the volume if (a) the diameter is exact, but the height is 20.1 meters; and (b) the height is exact, but the diameter is 5.1 meters.

A colony of bacteria grows exponentially and the colony's population is \(4,000\) at time \(t=0\) and \(6,500\) at time \(t=3 .\) How big is the population at time \(t=10 ?\)

A window consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches. Find the radius of the semicircle that will maximize the area of the window.

Find the area of the region between the two curves in each problem, and be sure to sketch each one. (We gave you only endpoints in one of them) The answers are in Chapter 19 . The curve \(x=y^{2}-4 y+2\) and the line \(x=y-2\).