Use calculus to find the radius that will maximize the volume of the cylinder if the surface area of the cylinder is held constant at 2 in². For this problem you will complete four steps.

Step 1: Identify the quantity to be optimized and the constraint.

What is the quantity to be optimized?

What is the constraint?

Answer choices:

r = 2/h

h = 2/r

r

2r² + 2rh

h

2r²+ 2rh = 2

r²h

r²h=2

Step 2: Express the quantity to be optimized as a function of a single variable.

Which function correctly expresses the volume as a function of a single variable (either h or r)?

Answer options:

A: V = (2/h)

B: V = r²- r³

C: V = 2/h

D: V = r - r³

Step 3: Find the endpoints and critical points of the unknown.

Given: V(r) = r - r³

In this scenario, what is the feasible domain for the value of r? Round your answer to two decimal places.

Apply the first derivative test to identify the critical points of the given function. Round your answer to two decimal points.

Step 4: Determine the dimensions that will maximize the volume of the cylinder.

Given: V(r) =r - r³, V'(r) - - 3r², r = 0 and r = 1 are the endpoints, and r = 0.58 is the critical point.

Determine the dimensions that will maximize the volume of the cylinder. Round your answers to two decimal places.

r=?

h = ?

Determine the maximum volume of a cylinder that has a surface are of 2 in². Round your answers to two decimal places.

V=?