Sphere

Sphere

Cylinder

Triangle Problems

Cone

Misc.

100

Air is being pumped into a spherical balloon at a rate of
5 cm³/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm

1/80π

100

The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. What is the rate of change of the volume of the cylinder, when the radius is 2 cm and the height is 3 cm?

2π/5

100

A police cruiser approaching a right-angled intersection from the North, is chasing a speeding car that has turned around the corner and is now headed East. When the cruiser is 0.6 miles North of the intersection and the car is 0.8 miles to the East, the police determine that the distance between them and the car is increasing at a rate of 20 mph. If the cruiser is moving at 60 mph at this instant, what is the speed of the car?

70 mph

100

A water tank in the shape of a right circular cone has a height of 10 feet. The top of the rim of the tank is a circle with a radius of 4 feet. If water is being pumped into the tank at the rate of 2 cubic feet/minute, what is the rate of change of the water level, when the depth is 5 feet?

1/2π ft/min

100

What variable do we always differentiate with respect to in a related rates problem? What does this variable represent?

t - Time

200

A small spherical meteorite enters the atmosphere. The volume of this meteorite decreases at a rate of 2 cm³/min. How fast is the Surface Area decreasing at this time when the diameter of the meteorite is 10 cm?
Hint: (SA of sphere= 4πr²)

-4/5

200

A right cylindrical tank is filled with water. The tank stands up and has a radius of 20 cm. At what rate does the height of the water in the tank drop when the water is being released at 25 cubic centimeters per second?

-1/16π

200

A ladder is 20 feet in length and leans against the side of a building. If the bottom of the ladder slides away from the building horizontally at a rate of 4 feet per second, at what rate is the ladder sliding down the building when the top of the ladder is 8 feet from the ground?

___
-√336/2

200

Water runs into a conical tank at the rate of 9 cubic feet per minute. The tank stands pointed downward and has a height of 10 feet and a base radius of 5 feet. How fast is the water level rising when the water is 6 feet deep?

1/π

200

List all of the steps when solving a Related Rates problem.

1) Draw and label a diagram
2) Write down what you're trying to find and what you know
3) Set up an equation that relates the variables
4) Differentiate both sides with respect to t
5) Substitute in values
6) Solve for the specific variable

500

Water is flowing at the rate of 6 m³/min from a reservoir shaped like a hemispherical bowl with a radius of 13 m. At what rate is the water level changing when the water is 8 m deep?
Hint: (R is V= (π/3)y²·(3R-y) )

-1/24π m/min

500

The radius of a right circular cylinder increases at a rate of 0.1 cm/sec, and the height decreases at a rate of 0.2 cm/sec. What is the rate of change of the volume of the cylinder, when the radius is 2 cm and the volume is 12π cubic cm/sec?

0.4π

500

Two people are walking on two different straight roads that meet at a right angle. Student A approaches that intersection at 2 m/sec and Student B is walking away from the intersection at 1 m/sec. At what rate is the angle, at B, changing when student A is 10 meters from the intersection and student B is 20 meters from the intersection. Transfer your answer into radians per second

18/π

500

Sand is being dumped through a funnel at a rate of 10 cubic feet/min and falls in a conical pile whose bottom radius is always exactly one-half of the height. How fast will the radius of the base change when the pile is 8 feet high?

10/128π

500

Some kids are trying to increase the size of a 6-inch deep tube with a machine that increases the tube's radius 1/1000th of an inch every three minutes. How rapidly is the tube's volume increasing when the diameter is 3.8 inches?

0.024 in³/min

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