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A right circular cylinder is inscribed in a sphere with radius R. Find the largest possible volume of such a cylinder.
Solution:
To visualize the problem, let's draw the figure first. Inscribed means inside and so a right circular cylinder is located inside the sphere.
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| Photo by Math Principles in Everyday Life |
Next, we have to find the dimensions of a right circular cylinder in order to get its volume. By labeling the figure further, we have
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| Photo by Math Principles in Everyday Life |
By Pythagorean Theorem,
We know that the volume of a right circular cylinder is
but h = 2a
Equate r2 on both sides of the equation,
Take the derivative of both sides of the equation with respect to a and equate it to zero because we want to maximize the volume of a right circular cylinder. Consider R as a constant in the equation.
Now, we can solve for r,
Therefore,
