91Ó°ÊÓ

Find \(D_{x} y\) by implicit differentiation. $$ 2 x^{3} y+3 x y^{3}=5 $$

A spherical balloon is being inflated so that its volume is increasing at the rate of \(5 \mathrm{ft}^{3} / \mathrm{min}\). At what rate is the diameter increasing when the diameter is \(12 \mathrm{ft}\) ?

A spherical snowball is being made so that its volume is increasing at the rate of \(8 \mathrm{ft}^{3} / \mathrm{min}\). Find the rate at which the radius is increasing when the snowball is \(4 \mathrm{ft}\) in diameter.

If water is being drained from a swimming pool and \(V\) gal is the volume of water in the pool \(t\) min after the draining starts, where \(V=250(40-t)^{2}\), find (a) the average rate at which the water leaves the pool during the first \(5 \mathrm{~min}\), and (b) how fast the water is flowing out of the pool 5 min after the draining starts.

In Exercises 9 through 12 , the motion of a particle is along a horizontal line according to the given equation of motion, where \(s \mathrm{ft}\) is the directed distance of the particle from a point \(O\) at \(t \mathrm{sec}\). The positive direction is to the right. Determine the intervals of time when the particle is moving to the right and when it is moving to the left. Also determine when the particle reverses its direction. Show the behavior of the motion by a figure similar to Fig. 3.2.2, choosing values of \(t\) at random but including the values of \(t\) when the particle reverses its direction. $$ s=t^{3}+3 t^{2}-9 t+4 $$

A trough is \(12 \mathrm{ft}\) long and its ends are in the form of inverted isosceles triangles having an altitude of \(3 \mathrm{ft}\) and a base of \(3 \mathrm{ft}\). Water is flowing into the trough at the rate of \(2 \mathrm{ft}^{3} / \mathrm{min}\). How fast is the water level rising when the water is 1.ft deep?

The motion of a particle is along a horizontal line according to the given equation of motion, where \(s \mathrm{ft}\) is the directed distance of the particle from a point \(O\) at \(t \mathrm{sec}\). The positive direction is to the right. Determine the intervals of time when the particle is moving to the right and when it is moving to the left. Also determine when the particle reverses its direction. Show the behavior of the motion by a figure similar to Fig. 3.2.2, choosing values of \(t\) at random but including the values of \(t\) when the particle reverses its direction. $$ s=2 t^{3}-3 t^{2}-12 t+8 $$

Boyle's law for the expansion of gas is \(P V=C\), where \(P\) is the number of pounds per square unit of pressure, \(V\) is the number of cubic units of volume of the gas, and \(C\) is a constant. At a certain instant the pressure is \(3000 \mathrm{lb} / \mathrm{ft}^{2}\), the volume is \(5 \mathrm{ft}^{3}\), and the volume is increasing at the rate of \(3 \mathrm{ft}^{3} / \mathrm{min}\). Find the rate of change of the pressure at this instant.

A man on a dock is pulling in a boat at the rate of \(50 \mathrm{ft} / \mathrm{min}\) by means of a rope attached to the boat at water level. If the man's hands are \(16 \mathrm{ft}\) above the water level, how fast is the boat approaching the dock when the amount of rope out is \(20 \mathrm{ft}\) ?

A rocket is fired vertically upward, and it is \(s \mathrm{ft}\) above the ground \(t \mathrm{sec}\) after being fired, where \(s=560 t-16 t^{2}\) and the positive direction is upward. Find (a) the velocity of the rocket 2 sec after being fired, and (b) how long it takes for the rocket to reach its maximum height.