Understanding the Volume of a Cone and Differentiation

The video tutorial explains how to find the rate of change of the height of a cone with respect to its radius, given the volume. It starts by introducing the formula for the volume of a cone and sets up the problem with specific values for the radius and height. The tutorial then demonstrates the differentiation process using the product rule to find dh/dr. Finally, it solves the equation and calculates the result using the given values.

What is the formula for the volume of a right circular cone?

• V = 1/2 πr²h

• V = 2πrh

• V = 1/3 πr²h

• V = πr²h

What is the given volume of the cone in the problem?

• 3π cubic centimeters

• 10π cubic centimeters

• 15π cubic centimeters

• 5π cubic centimeters

What is the goal of the problem in terms of differentiation?

• Find the derivative of height with respect to radius

• Find the derivative of radius with respect to height

• Find the derivative of volume with respect to height

• Find the derivative of radius with respect to volume

Which rule is applied to differentiate the equation with respect to radius?

• Chain Rule

• Product Rule

• Quotient Rule

• Power Rule

What is the derivative of 1/3 πr² with respect to r?

• 2/3 πr

• 3/2 πr

• 1/3 πr

• πr

What is the simplified expression for dh/dr after solving the equation?

• -2h/r

• 2h/r

• h/2r

• -h/2r

What is the first step in solving for dh/dr after applying the product rule?

• Divide both sides by π

• Subtract 2/3 πrh from both sides

• Multiply both sides by 3

• Add 2/3 πrh to both sides

What happens to the π terms when simplifying the expression for dh/dr?

• They cancel out

• They are squared

• They are halved

• They double

What is the value of dh/dr when r = 3 cm and h = 15 cm?

• -10

• 5

• 10

• -5

What is the final step in calculating dh/dr for given values of r and h?

• Add the values

• Substitute the values and simplify

• Divide by h

• Multiply by r