Question 244866
The volume of a cylinder, using "x" for the height and "r" for the radius of the circular bases, is:
{{{V = pi*r^2*x}}}
The problem is to express r in terms of x and R. (It will not be possible, given the information you provided, to express r just in terms of x.)<br>
The keys to this solution are:<ul><li>The center of the sphere and the center of the cylinder will be the same point.</li><li>The distance from the center of the cylinder to the center of either of the two circular bases will be one half of the height of the cylinder.<li>The points of the two circular bases of the cylinder will be on the sphere, too.</li></ul>
Once you understand the above, you will understand the following diagram:
{{{drawing(300, 300, 0, 4, 0, 4, line(0, 1, 0, 3), line(0, 3, 3, 3), line(0, 1, 3, 3), locate(0.1, 2.3, x/2), locate(1.4, 2.9, r), locate(1.5, 1.8, R), locate(0.1, 1, A), locate(0.1, 3.2, B))}}}
A = the center of the sphere
B = the center of one of the circular bases of the cylinder
r = radius of the circular base of the cylinder
R = radius of the sphere
x = height of the cylinder<br>
This is a right triangle so we can use the Pythagorean Theorem:
{{{(x/2)^2 + r^2 = R^2}}}
Simplifying we get:
{{{x^2/4 + r^2 = R^2}}}
Solving for r:
{{{r^2 = R^2 - x^2/4}}}
{{{r = sqrt(R^2 - x^2/4)}}} (discarding the negative square root because radii are not negative)<br>
Now we can subsitute for r in the Volume formula:
{{{V = pi*(sqrt(R^2 - x^2/4))^2*x}}}
Simplifying we get:
{{{V = pi*(R^2 - x^2/4)*x}}}
{{{V = pi*x*R^2 - pi*x^3/4}}}