Related Rates

Front

What is the formula for the volume of a cylinder?

Back

V = πr²h, where V is the volume, r is the radius, and h is the height.

Front

How do you find the rate of change of the height of water in a cylindrical tank?

Back

Use the formula dV/dt = πr²(dh/dt), where dV/dt is the rate of volume change, r is the radius, and dh/dt is the rate of height change.

Front

What is the relationship between the radius and height of an inverted cone?

Back

For an inverted cone, the volume V = (1/3)πr²h, and the radius is often related to the height by a constant ratio.

Front

How do you apply related rates in problems involving ladders?

Back

Use the Pythagorean theorem to relate the height and base distance of the ladder, then differentiate with respect to time.

Front

What is the formula for the area of a circle?

Back

A = πr², where A is the area and r is the radius.

Front

How do you find the rate of change of the area of a circle as the radius changes?

Back

Use the formula dA/dt = 2πr(dr/dt), where dA/dt is the rate of area change, r is the radius, and dr/dt is the rate of radius change.

Front

What is the significance of the derivative in related rates problems?

Back

The derivative represents the rate of change of a quantity with respect to time, allowing us to relate different rates in a problem.

Front

How do you set up a related rates problem?

Back

Identify the quantities involved, establish relationships between them, differentiate with respect to time, and solve for the desired rate.

Front

What is the relationship between the height of a water tank and the volume of water it contains?

Back

The volume of water in a tank is directly related to the height of the water, given a constant cross-sectional area.

Front

How do you determine the rate at which a fish is approaching the dock?

Back

Use the Pythagorean theorem to relate the distance of the fish from the dock and the length of the fishing line, then differentiate.