Solve the problem.
-Assuming that a cylindrical container can be mailed only if the sum of its height and circumference do not exceed 270 centimeters, what are the dimensions of the cylinder with the largest volume that can be mailed?
A) Height 90 centimeters and radius $90 / \pi$ centimeters
B) Height 180 centimeters and radius 90 centimeters
C) Height 270 centimeters and radius $90 / \pi$ centimeters
D) Height 90 centimeters and radius $270 / \pi$ centimeters

Question

Quizplus · Accepted Answer

The sum of the height and circumference of a cylinder with radius r and height h is 2πr + h.  To maximize the volume, we need to maximize πr^2h subject to the constraint 2πr + h ≤ 270.  Using the constraint, we can solve for h in terms of r: h = 270 - 2πr.  Substituting this into the volume equation, we get V = πr^2(270 - 2πr) = 270πr^2 - 2π^2r^3.  To find the maximum, we take the derivative and set it equal to 0: dV/dr = 540πr - 6π^2r^2 = 0.  Solving for r, we get r = 90/π.  Substituting this back into the equation for h, we get h = 90.  Therefore, the dimensions of the cylinder with the largest volume that can be mailed are height 90 centimeters and radius 90/π centimeters.

Solve the Problem 90/π90 / \pi90/π Centimeters B) Height 180 Centimeters and Radius 90 Centimeters C)

Solve the Problem $90 / \pi$ Centimeters
B) Height 180 Centimeters and Radius 90 Centimeters
C)