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 180 centimeters and radius 90 centimeters
B) Height 90 centimeters and radius $90 / \pi$ 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 of the cylinder, we need to maximize the product of the base area and the height. The base area is πr^2, so the volume is V = πr^2h. We can use the constraint 2πr + h ≤ 270 to solve for h in terms of r: h ≤ 270 - 2πr. Substituting this into the expression for the volume, we get V ≤ πr^2(270 - 2πr). To maximize this expression, we take the derivative with respect to r and set it equal to 0:dV/dr = 2πr(270 - 2πr) - πr^2(2π) = 0540πr - 4π^2r^2 - 2π^2r^2 = 04π^2r^2 - 540πr = 04πr(πr - 135) = 0r = 0 or r = 135/πSince r cannot be 0, we have r = 135/π. Substituting this back into the expression for h, we get h ≤ 270 - 2π(135/π) = 90. Therefore, the dimensions of the cylinder with the largest volume that can be mailed are height 90 centimeters and radius 135/π centimeters, which is choice B.

Solve the Problem 90/π90 / \pi90/π Centimeters C) Height 270 Centimeters and Radius

Solve the Problem $90 / \pi$ Centimeters
C) Height 270 Centimeters and Radius