Question

The volume of a cylinder is given by the formula V = πr2 h. A company is creating cylindrical packaging for a new product. The height of the package is set at 5 inches. The volume can be no greater than 250 in3 . a. Write the inverse function expressing r as a function of V. b. Determine a reasonable domain and range of the inverse function.

a) r = (V/πh)^0.5
b r = (V/π)^0.5
c) r = (V/πh)^2
d)r = (V/π)^2

Answer 1

The inverse function expressing radius as a function of volume for a cylinder with height 5 inches is r = √(V / (π * 5)). The reasonable domain is 0 ≤ V ≤ 250 in³, and the range is from near 0 up to the radius of a cylinder with a volume of 250 in³.

Step-by-step explanation:

The formula for the volume of a cylinder is given by V = πr²h, where V is the volume, r is the radius of the cylinder's base, and h is the height. To express the radius r as a function of the volume V, we rearrange the given formula to solve for r:

V = πr²h
→ r² = V / (πh)
→ r = √(V / (πh))

So, given the height of the cylinder is set to 5 inches and the volume is no greater than 250 in³, the inverse function is:

r = √(V / (π * 5))

The correct choice from the provided options is (a) r = (V/πh)^0.5.

The reasonable domain of this function would be 0 ≤ V ≤ 250, as the volume cannot be negative and must not exceed 250 in³. The reasonable range for the radius would be from the radius of a cylinder with near zero volume (approaching 0) up to the radius of a cylinder with the maximum allowable volume.