SOLUTION: Given a sphere of radius R, find the radius r and altitude 2h of the right circular cylinder with largest lateral surface area that can be inscribed in the sphere.
I think I was
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I think I was
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Question 684171: Given a sphere of radius R, find the radius r and altitude 2h of the right circular cylinder with largest lateral surface area that can be inscribed in the sphere.
I think I was able to calculate the function but I am not sure if it is correct. Also, please include all steps to the solution. This is a optimization problem. Answer by Edwin McCravy(20054) (Show Source):
This is a cross section cut through the center of the sphere:
The lateral area of circular cylinder is
By the Pythagorean theorem (refer to the drawing): , so we substitute:
Since square roots are difficult to work with, let's square both sides:
The trick here is that if we maximize the SQUARE of the lateral area,
we will also have maximized the lateral area. So we let S = AČ
We set that equal to zero:
0
0
Divide through by constant 0
r=0; RČ - 2rČ = 0
(min, area = 0) -2rČ = -RČ
rČ =
r =
So the radius of the cylinder which has maximum
surface area is
Since h =
h =
h =
h =
height = 2h =
Edwin
