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) (Show Source): You can put this solution on YOUR website!
r = radius
h = height
SA = surface area of the cylinder
Plug in the given surface area of 10 square units. From here we isolate h.
Divide both sides by 2.
We'll use this so we can eliminate the variable h in the next equation below.
V = volume of the cylinder
Plug in the equation we solved previously
We end up with the volume in terms of one single variable.
Let's consider the function
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
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