SOLUTION: For a cylinder with a surface area of 10 , what is the maximum volume that it can have? Round your answer to the nearest 4 decimal places. Recall that the volume of a cylinder

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Question 1204616: For a cylinder with a surface area of 10
, what is the maximum volume that it can have? Round your answer to the nearest 4 decimal places.
Recall that the volume of a cylinder is πr2h
and the surface area is 2πrh+2πr2
where r
is the radius and h
is the height.
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

r = radius
h = height

SA = surface area of the cylinder
SA+=+2%2Api%2Ar%2Ah+%2B+2%2Api%2Ar%5E2

10+=+2%2Api%2Ar%2Ah+%2B+2%2Api%2Ar%5E2 Plug in the given surface area of 10 square units. From here we isolate h.

10+=+2%2A%28pi%2Ar%2Ah+%2B+pi%2Ar%5E2%29

10%2F2+=+pi%2Ar%2Ah+%2B+pi%2Ar%5E2 Divide both sides by 2.

5+=+pi%2Ar%2Ah+%2B+pi%2Ar%5E2

5-pi%2Ar%5E2+=+pi%2Ar%2Ah

h+=+%285+-+pi%2Ar%5E2%29%2F%28pi%2Ar%29
We'll use this so we can eliminate the variable h in the next equation below.


V = volume of the cylinder
V+=+pi%2Ar%5E2%2Ah

V+=+pi%2Ar%5E2%2A%28%285+-+pi%2Ar%5E2%29%2F%28pi%2Ar%29%29 Plug in the equation we solved previously

V+=+r%2A%285+-+pi%2Ar%5E2%29

V+=+5r+-+pi%2Ar%5E3 We end up with the volume in terms of one single variable.

Let's consider the function f%28x%29+=+5x+-+pi%2Ax%5E3
x = r = radius
f(x) = V = volume

Domain: x > 0
Range: f(x) > 0
We focus on the upper right quadrant (aka Q1)

Use a graphing calculator, or derivatives (if you are in a calculus class), to find the highest point in Q1 occurs at the approximate location of (0.728366, 2.427885)
GeoGebra and Desmos are two graphing options that I recommend.

Here is the link to the interactive Desmos graph
https://www.desmos.com/calculator/pzk6yfzxpd
Click on the highest point to have its coordinates show up. You may have to click twice.

Therefore a radius of approximately 0.728366 units leads to the max cylinder volume of approximately 2.427885 cubic units. This applies only when the surface area is 10 square units.
A real world application is that you have 10 square units of material, and the goal is to get the most storage space out of the cylinder.


Answer: 2.4279 cubic units