Problem 42 - Essential Calculus

Question

A sphere of radius \(r\) is cut by a plane \(h(h

Step-by-Step Solution

Answer

The volume of the spherical segment above the plane is given by \(V = \frac{4}{3}\pi (r^3 - (\sqrt{{r^2 - h^2}})^3)\).

Step 1: Find radius of the top circle

First, you'll need to figure out the radius of the circle that is formed by the intersection of the plane with the sphere. You can do this by creating a right triangle with the original radius \(r\), height above the equator \(h\), and this radius \(r_1\). Then, you can use the Pythagorean theorem to find \(r_1 = \sqrt{{r^2 - h^2}}\).

Step 2: Find the volume of the smaller sphere

The volume of the smaller sphere, which is beneath the plane, can be found using the formula for the volume of a sphere \(\frac{4}{3}\pi r_1^3\).

Step 3: Find the volume of the bigger sphere

Using the same formula, you can find the volume of the bigger sphere with the radius \(r\), which is \(\frac{4}{3}\pi r^3\).

Step 4: Subtract the volumes

Subtract the volume of the smaller sphere from that of the larger sphere. This will give the volume of the spherical segment: \(V = \frac{4}{3}\pi r^3 - \frac{4}{3}\pi r_1^3\). Simplify the equation by factoring \(\frac{4}{3}\pi\) out: \(V = \frac{4}{3}\pi (r^3 - r_1^3)\).

Step 5: Substitute \(r_1\) into the volume formula

Finally, substitute \(r_1\) from step 1 to get the volume of the spherical segment in terms of \(r\) and \(h\): \(V = \frac{4}{3}\pi (r^3 - (\sqrt{{r^2 - h^2}})^3)\).