Solve. The volume of a cone varies jointly as its height and...

Solve.
The volume of a cone varies jointly as its height and the square of its radius. If the volume of a cone is $32 \pi$ cubic inches when the radius is 4 inches and the height is 6 inches, find the volume of a cone when the radius is 3 inches and the height is 5 inches.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Joint Variation

This concept refers to a situation where one quantity is directly proportional to the product of two or more other quantities. In the context of the problem, the volume of the cone varies directly as its height and as the square of its radius, meaning that any change in the height or radius will affect the volume in a predictable way according to a constant multiplier.

Constant of Proportionality

The constant of proportionality is the factor that relates the combined variables (in this case, the square of the radius and the height) to the actual value of the volume. Determining this constant using given values allows for the establishment of an equation that can be used to find the volume under different conditions.

Substitution and Solving Equations

Substitution involves replacing the variables with their known values to simplify and solve for the unknown constant or quantity. Once the constant is found, it can be used in the joint variation formula to compute the desired volume when new values for the radius and height are given.

Transcript

Please give Ace some feedback