Looking at a right circular cylinder inside of a sphere, if we look at a picture, we can see that the radius of the cylinder plus half the height of the cylinder, the radius squared of the cylinder plus half the height squared of the cylinder is going to equal the radius squared of the sphere. So if we put in some variables, let's let A be the radius of the cylinder, B be half the height of the cylinder and R be the radius of the sphere, we can see that a ^2 + b ^2 would equal R-squared. And this problem R is given a 70, so a ^2 + b ^2 = 70 ^2. If I solved for the radius of the cylinder, I'd have a squared equaling 4900 -, b ^2. Now the volume of a right circular cylinder is π R-squared H so volume equal π. The radius squared was that 4900 -, b ^2 times the height of the cylinder, which is going to be 2B because remember B was half of the whole height. If we distribute things out, we get volume equaling 9800 Pi b -, 2π B cubed. Taking the first derivative and setting it equal to 0, we get 0 equaling 9800 Pi -6 Pi b ^2. Solving for B, we get 70 square roots of 3 / 3. Now it didn't ask for B, it asked for the height and the height was 2B. So if 2B equal the height, we're just going to take the B and multiply by two and we get approximately 80.83. Once I get my BI, can substitute it and define my A. So my A is the square root of the 49,000 minus the b ^2, b ^2 being 4900 / 3. So A is sqrt 9800 / 3 or 70 square roots of 6 / 3. That's approximately 57.15. The next part asked for the volume, so the volume is just Pi r ^2 H when we plug those two numbers in. I did not use the rounding approximate because of rounding errors. I used the exact and I get 82951 O .91.