Chapter 25

Find the instantaneous velocity and acceleration at the given time for the straightline motion described by each equation, where \(s\) is in centimeters and \(t\) is in seconds. In this exercise assume that the integers in the given equations are exact numbers and give approximate answers to three significant digits. $$s=32 t-8 t^{2} \quad \text { at } t=2.00$$

The temperature \(T\) inside a certain furnace is described by the equation \(T=55.6 t^{2}+28.2 t+44.8^{\circ} \mathrm{F},\) where \(t\) is the elapsed time in hours. Find the time rate of change of the temperature at \(t=2.00 \mathrm{h}\)

Separate the number 10 into two parts such that their product will be a maximum.

Find the instantaneous velocity and acceleration at the given time for the straightline motion described by each equation, where \(s\) is in centimeters and \(t\) is in seconds. In this exercise assume that the integers in the given equations are exact numbers and give approximate answers to three significant digits. $$s=6 t^{2}-2 t^{3} \quad \text { at } t=1.00$$

The pressure \(p\) in a tank varies with time according to the function \(p=34.6 t^{3}-44.5 t \mathrm{lb} / \mathrm{in.}^{2},\) where \(t\) is in minutes. What is the time rate of change of pressure at \(t=5.50\) min?

A boat is fastened to a rope that is wound around a winch \(20.0 \mathrm{ft}\) above the level at which the rope is attached to the boat. The boat is drifting away at the horizontal rate of \(8.00 \mathrm{ft} / \mathrm{s}\). How fast is the rope increasing in length when 30.0 feet of rope is out?

The charge \(q\) (in coulombs) through a \(4.82-\Omega\) resistor varies with time according to the function \(q=3.48 t^{2}-1.64 t .\) Write an expression for the instantaneous current through the resistor.

A boat with its anchor on the bottom at a depth of \(40.0 \mathrm{m}\) is drifting away from the anchor at \(4.00 \mathrm{m} / \mathrm{s}\), while the anchor cable slips from the boat at water level. At what rate is the cable leaving the boat when 50.0 meters of cable is out? Assume that the cable is straight.

Find the instantaneous velocity and acceleration at the given time for the straightline motion described by each equation, where \(s\) is in centimeters and \(t\) is in seconds. In this exercise assume that the integers in the given equations are exact numbers and give approximate answers to three significant digits. $$s=120 t-16 t^{2} \quad \text { at } t=4.00$$

A kite is at a constant height of \(120 \mathrm{ft}\) and moves horizontally, at \(4.00 \mathrm{mi} / \mathrm{h}\) in a straight line away from the person holding the cord. Assuming that the cord remains straight, how fast is the cord being paid out when its length is \(130 \mathrm{ft} ?\)

Find the instantaneous velocity and acceleration at the given time for the straightline motion described by each equation, where \(s\) is in centimeters and \(t\) is in seconds. In this exercise assume that the integers in the given equations are exact numbers and give approximate answers to three significant digits. $$s=3 t-t^{4}-8 \quad \text { at } t=1.00$$

The distance in feet traveled in time \(t\) seconds by a point moving in a straight line is given by the formula \(s=40 t+16 t^{2} .\) Find the velocity and the acceleration at the end of \(2.00 \mathrm{s}.\)

The charge \(q\) (in coulombs) at a resistor varies with time according to the function \(q=22.4 t+41.6 t^{3} .\) Write an expression for the instantaneous current through the resistor, and evaluate it at \(2.50 \mathrm{s}\)

If the distance traveled by a ball rolling down an incline in \(t\) seconds is \(s\) feet, where \(s=6 t^{2},\) find its speed when \(t=5.00 \mathrm{s}.\)

The voltage applied to a \(33.5-\mu \mathrm{F}\) capacitor is \(v=6.27 t^{2}-15.3 t+52.2 \mathrm{V}\) Find the current at \(t=5.50 \mathrm{s}\)

The height \(s\) in feet reached by a ball \(t\) seconds after being thrown vertically upward at \(320 \mathrm{ft} / \mathrm{s}\) is given by \(s=320 t-16 t^{2} .\) Find (a) the greatest height reached by the ball and (b) the velocity with which it reaches the ground.

The voltage applied to a \(1.25-\mu \mathrm{F}\) capacitor is \(v=3.17+28.3 t+29.4 t^{2} \mathrm{V}\) Find the current at \(t=33.2 \mathrm{s}\)

A lamp is located on the ground \(30.0 \mathrm{ft}\) from a building. A person \(6.00 \mathrm{ft}\) tall walks from the light toward the building at a rate of \(5.00 \mathrm{ft} / \mathrm{s} .\) Find the rate at which the person's shadow on the wall is shortening when the person is \(15.0 \mathrm{ft}\) from the building.

Find the area of the greatest rectangle that has a perimeter of 20.0 in.

A rocket was fired straight upward so that its height in feet after \(t\) seconds was \(s=2000 t-16 t^{2}.\) (a) What was its initial velocity? (b) What was its greatest height? (c) What was its velocity at the end of \(10.0 \mathrm{s} ?\)

The current in a \(1.44-\mathrm{H}\) inductor is given by \(i=5.22 t^{2}-4.02 t .\) Find the voltage across the inductor at \(t=2.00 \mathrm{s}\)

The height \(h\) in kilometers to which a balloon will rise in \(t\) minutes is given by the formula $$h=\frac{10 t}{\sqrt{4000+t^{2}}}$$ At what rate is the balloon rising at the end of 30.0 min?

The current in a \(8.75-\mathrm{H}\) inductor is given by \(i=8.22+5.83 t^{3} .\) Find the voltage across the inductor at \(t=25.0 \mathrm{s}\)

A square sheet of metal 10.0 in. on a side is expanded by increasing its temperature so that each side of the square increases 0.00500 in./s. At what rate is the area of the square increasing at \(20.0 \mathrm{s} ?\)

The air in a certain cylinder is at a pressure of \(25.5 \mathrm{lb} / \mathrm{in.}^{2}\) when its volume is 146 in. \(^{3} .\) Find the rate of change of the pressure with respect to volume as the piston descends farther. Use Boyle's law, \(p v=k\)

A circular plate in a furnace is expanding so that its radius is changing \(0.010 \mathrm{cm} / \mathrm{s} .\) How fast is the area of one face changing when the radius is \(5.00 \mathrm{cm} ?\)

A point moves along the curve \(y=2 x^{3}-3 x^{2}-2 \mathrm{cm}.\) (a) Find the direction of travel at \(x=1.50 \mathrm{cm} .\) (b) If the speed of the point along the curve is \(3.75 \mathrm{cm} / \mathrm{s}\), find the \(x\) and \(y\) \( \mathrm{com}\) ponents of the velocity when \(x=1.50 \mathrm{cm} .\)

A certain light source produces an illumination of 655 lux on a surface at a distance of \(2.75 \mathrm{m} .\) Find the rate of change of illumination with respect to distance, and evaluate it at \(2.75 \mathrm{m} .\) Use the inverse square law, \(I=k / d^{2}\)

The volume of a cube is increasing at 10.0 in. \(^{3} /\) min. At the instant when its volume is 125 in. \(^{3},\) what is the rate of change of its edge?

A point moves along the curve \(y=x^{4}+x^{2}\) in. (a) Find the direction of travel at \(x=2.55\) in. (b) If the speed of the point along the curve is 1.25 in./s, find the \(x\) and \(y\) components of the velocity when \(x=2.55\) in.

A spherical balloon starts to shrink as the gas escapes. Find the rate of change of its volume with respect to its radius when the radius is \(1.00 \mathrm{m} .\left(V=\frac{4}{3} \pi r^{3}\right)\)

The edge of an expanding cube is changing at the rate of 0.00300 in./s. Find the rate of change of its volume when its edge is 5.00 in. long.

A point moves along a curve such that its horizontal displacement is \(x=3 t^{2}+5 t \mathrm{cm}\) and its vertical displacement is \(y=13-3 t^{2} \mathrm{cm} .\) Find the horizontal and vertical components of the instantaneous velocity and acceleration at \(t=4.55 \mathrm{s}.\)

At some instant the diameter \(x\) of a cylinder (Fig. \(25-25\) ) is 10.0 in. and is increasing at a rate of 1.00 in./min. At that same instant, the height \(y\) is 20.0 in. and is decreasing at a rate \((d y / d t)\) such that the volume is not changing \((d V / d t=0) .\) Find \(d y / d t\)

The period (in seconds) for a pendulum of length \(L\) in. to complete one oscillation is equal to \(P=0.324 \sqrt{L} .\) Find the rate of change of the period with respect to length when the length is 9.00 in.

Water is running from a vertical cylindrical tank \(3.00 \mathrm{m}\) in diameter at the rate of \(3 \pi \sqrt{h} \mathrm{m}^{3} / \mathrm{min},\) where \(h\) is the depth of the water in the tank. How fast is the surface of the water falling when \(h=9.00 \mathrm{m} ?\)

The temperature \(T\) at a distance \(x\) in. from the end of a certain heated bar is given by \(T=2.24 x^{3}+1.85 x+95.4\left(^{\circ} \mathrm{F}\right) .\) Find the rate of change of temperature with respect to distance, which is called the temperature gradient, at a point 3.75 in. from the end.

A point has horizontal and vertical displacements (in \(\mathrm{cm}\) ) of \(x=4-2 t^{2}\) and \(y=5 t^{2}+3,\) respectively. (a) Find the \(x\) and \(y\) components of the velocity and acceleration at \(t=2.75 \mathrm{s}.\) (b) Find the magnitude and direction of the resultant velocity.

Sand poured on the ground at the rate of \(3.00 \mathrm{m}^{3} / \mathrm{min}\) forms a conical pile whose height is one-third the diameter of its base. How fast is the altitude of the pile increasing when the radius of its base is \(2.00 \mathrm{m} ?\)

A projectile is launched at an angle of \(27.0^{\circ}\) to the horizontal with an initial velocity of \(1260 \mathrm{ft} / \mathrm{s}\). Find (a) the horizontal and vertical positions of the projectile and (b) the horizontal and vertical velocities, after 3.25 s.

The angular displacement of a rotating body is given by \(\theta=44.8 t^{3}+29.3 t^{2}+81.5\) rad. Find the angular velocity at \(t=4.25 \mathrm{s}.\)

The adiabatic law for the expansion of air is \(p v^{1.4}=C .\) If at a given time the volume is observed to be \(10.0 \mathrm{ft}^{3},\) and the pressure is \(50.0 \mathrm{lb} / \mathrm{in} .^{2},\) at what rate is the pressure changing if the volume is decreasing \(1.00 \mathrm{ft}^{3} / \mathrm{s} ?\)

The angular displacement of a rotating body is given by \(\theta=184+271 t^{3}\) rad. Find (a) the angular velocity and (b) the angular acceleration, at \(t=1.25 \mathrm{s}.\)

Two trains start from the same point at the same time, one going east at a rate of \(40.0 \mathrm{mi} / \mathrm{h}\) and the other going south at \(60.0 \mathrm{mi} / \mathrm{h} .\) Find the rate at which they are separating after \(1.00 \mathrm{h}\) of travel.

The angular displacement of a rotating body is given by \(\theta=2.84 t^{3}-7.25\) rad. Find (a) the angular velocity and (b) the angular acceleration, at \(t=4.82 \mathrm{s}.\)

An airplane leaves a field at noon and flies east at \(100 \mathrm{km} / \mathrm{h}\). A second airplane leaves the same field at 1 P.M. and flies south at \(150 \mathrm{km} / \mathrm{h}\). How fast are the airplanes separating at 2 P.M.?

The speed \(v \mathrm{ft} / \mathrm{s}\) of a certain bullet passing through wood is given by \(v=500 \sqrt{1-3 x},\) where \(x\) is the depth in feet. Find the rate at which the speed is decreasing after the bullet has penetrated 3.00 in. (Hint: When substituting for \(d x / d t, \text { simply use the given expression for } v .)\)

As a man walks a distance \(x \text { along a board (Fig. } 25-27),\) he sinks a distance of \(y\) in., where $$ y=\frac{P x^{3}}{3 E I} $$ Here, \(P\) is the person's weight, 165 lb; \(E\) is the modulus of elasticity of the material in the board, \(1,320,0001 b /\) in. \(^{2} ;\) and \(I\) is the modulus of elasticity of the cross section, 10.9 in. \(^{4} .\) If he moves at the rate of 25.0 in./s, how fast is he sinking when \(x=75.0\) in.?

A stone dropped into a calm lake causes a series of circular ripples. The radius of the outer one increases at \(2.00 \mathrm{ft} / \mathrm{s} .\) How rapidly is the disturbed area changing at the end of 3.00 s?

The power delivered to a load by a \(30-\mathrm{V}\) source of internal resistance \(2 \Omega\) is \(30 i-2 i^{2} \mathrm{W},\) where \(i\) is the current in amperes. For what current will this source deliver the maximum power?