Cylinder Volume and Rate of Change

The video tutorial explains how to calculate the rate at which the height of water in a leaking right cylinder changes. It begins by introducing the problem and the volume formula for a cylinder. The tutorial then demonstrates how to use derivatives to find the rate of change of the water's height, given the rate at which the cylinder is leaking. The final calculation shows that the height decreases at a rate of 0.11 feet per second when the water height is 4 feet.

What is the rate at which the cylinder is leaking water?

• 2.5 cubic feet per second

• 5.2 cubic feet per second

• 3.1 cubic feet per second

• 4.0 cubic feet per second

What is the formula for the volume of a right cylinder?

• V = 2πr²h

• V = πr² + h

• V = 2πrh

• V = πr²h

What is the radius of the cylinder in the problem?

• 2 feet

• 3 feet

• 4 feet

• 5 feet

When the height of the water is 4 feet, what is the radius of the water in the cylinder?

• 2 feet

• 5 feet

• 3 feet

• 4 feet

What is the derivative of the volume with respect to time?

• dv/dt = 3.1π

• dv/dt = πr²(dh/dt)

• dv/dt = 9π(dh/dt)

• dv/dt = 9πh

How do you solve for dh/dt in the derived equation?

• Multiply both sides by 9π

• Add 3.1 to both sides

• Subtract 3.1 from both sides

• Divide both sides by 9π

What is the calculated rate of change of the height of the water?

• 0.11 feet per second

• 0.21 feet per second

• -0.11 feet per second

• -0.21 feet per second

What does a negative rate of change of height indicate?

• The height is increasing

• The height is decreasing

• The height is constant

• The height is fluctuating

At what height of the water is the rate of change calculated?

• 6 feet

• 3 feet

• 4 feet

• 5 feet

What are the units of the rate of change of height?

• Meters per second

• Inches per second

• Feet per second

• Cubic feet per second