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The radius of a sphere is increasing at a rate of 9 cm/sec. Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.
$\frac{1}{2\sqrt{\pi}}$ cm
The base of a triangle is shrinking at a rate of 1 cm/min and the height of the triangle is increasing at a rate of 5 cm/min. Find the rate at which the area of the triangle changes when the height is 22 cm and the base is 10 cm.
A triangle has two constant sides of length 3 ft and 5 ft. The angle between these two sides is increasing at a rate of 0.1 rad/sec. Find the rate at which the area of the triangle is changing when the angle between the two sides is $\pi \text{/}6.$
The area is increasing at a rate $\frac{\left(3\sqrt{3}\right)}{8}\phantom{\rule{0.2em}{0ex}}{\text{ft}}^{\phantom{\rule{0.2em}{0ex}}}\text{/sec.}$
A triangle has a height that is increasing at a rate of 2 cm/sec and its area is increasing at a rate of 4 cm ^{2} /sec. Find the rate at which the base of the triangle is changing when the height of the triangle is 4 cm and the area is 20 cm ^{2} .
For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft.
How fast does the depth of the water change when the water is 10 ft high if the cone leaks water at a rate of 10 ft ^{3} /min?
The depth of the water decreases at $\frac{128}{125\pi}$ ft/min.
Find the rate at which the surface area of the water changes when the water is 10 ft high if the cone leaks water at a rate of 10 ft ^{3} /min.
If the water level is decreasing at a rate of 3 in./min when the depth of the water is 8 ft, determine the rate at which water is leaking out of the cone.
The volume is decreasing at a rate of $\frac{\left(25\pi \right)}{16}{\text{ft}}^{3}\text{/min}.$
A vertical cylinder is leaking water at a rate of 1 ft ^{3} /sec. If the cylinder has a height of 10 ft and a radius of 1 ft, at what rate is the height of the water changing when the height is 6 ft?
A cylinder is leaking water but you are unable to determine at what rate. The cylinder has a height of 2 m and a radius of 2 m. Find the rate at which the water is leaking out of the cylinder if the rate at which the height is decreasing is 10 cm/min when the height is 1 m.
The water flows out at rate $\frac{\left(2\pi \right)}{5}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\phantom{\rule{0.2em}{0ex}}}\text{/min.}$
A trough has ends shaped like isosceles triangles, with width 3 m and height 4 m, and the trough is 10 m long. Water is being pumped into the trough at a rate of $5\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}\text{/min}.$ At what rate does the height of the water change when the water is 1 m deep?
A tank is shaped like an upside-down square pyramid, with base of 4 m by 4 m and a height of 12 m (see the following figure). How fast does the height increase when the water is 2 m deep if water is being pumped in at a rate of $\frac{2}{3}$ m/sec?
$\frac{3}{2}$ m/sec
For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft ^{3} /min. The radius of the pool is 10 ft.
Find the rate at which the depth of the water is changing when the water has a depth of 5 ft.
Find the rate at which the depth of the water is changing when the water has a depth of 1 ft.
$\frac{25}{19\pi}$ ft/min
If the height is increasing at a rate of 1 in./sec when the depth of the water is 2 ft, find the rate at which water is being pumped in.
Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of 10 ft ^{3} /min. The radius of the cone base is three times the height of the cone. Find the rate at which the height of the gravel changes when the pile has a height of 5 ft.
$\frac{2}{45\pi}$ ft/min
Using a similar setup from the preceding problem, find the rate at which the gravel is being unloaded if the pile is 5 ft high and the height is increasing at a rate of 4 in./min.
For the following exercises, draw the situations and solve the related-rate problems.
You are stationary on the ground and are watching a bird fly horizontally at a rate of $10$ m/sec. The bird is located 40 m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is 9 m?
The angle decreases at $\frac{400}{1681}\phantom{\rule{0.2em}{0ex}}\text{rad/sec}.$
You stand 40 ft from a bottle rocket on the ground and watch as it takes off vertically into the air at a rate of 20 ft/sec. Find the rate at which the angle of elevation changes when the rocket is 30 ft in the air.
A lighthouse, L , is on an island 4 mi away from the closest point, P , on the beach (see the following image). If the lighthouse light rotates clockwise at a constant rate of 10 revolutions/min, how fast does the beam of light move across the beach 2 mi away from the closest point on the beach?
$100\pi \text{/}\text{min}$
Using the same setup as the previous problem, determine at what rate the beam of light moves across the beach 1 mi away from the closest point on the beach.
You are walking to a bus stop at a right-angle corner. You move north at a rate of 2 m/sec and are 20 m south of the intersection. The bus travels west at a rate of 10 m/sec away from the intersection – you have missed the bus! What is the rate at which the angle between you and the bus is changing when you are 20 m south of the intersection and the bus is 10 m west of the intersection?
The angle is changing at a rate of $\frac{21}{25}\phantom{\rule{0.2em}{0ex}}\text{rad/sec}.$
For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft.
[T] A batter hits a ball toward third base at 75 ft/sec and runs toward first base at a rate of 24 ft/sec. At what rate does the distance between the ball and the batter change when 2 sec have passed?
[T] A batter hits a ball toward second base at 80 ft/sec and runs toward first base at a rate of 30 ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? ( Hint : Recall the law of cosines.)
The distance is increasing at a rate of $62.50$ ft/sec.
[T] A batter hits the ball and runs toward first base at a speed of 22 ft/sec. At what rate does the distance between the runner and second base change when the runner has run 30 ft?
[T] Runners start at first and second base. When the baseball is hit, the runner at first base runs at a speed of 18 ft/sec toward second base and the runner at second base runs at a speed of 20 ft/sec toward third base. How fast is the distance between runners changing 1 sec after the ball is hit?
The distance is decreasing at a rate of $11.99$ ft/sec.
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