Draw a perpendicular from A to line BC. I labeled the intersection D. Let x = AD and y = BD. By the pythagorean theorem , we know that
x² + y² = 25² and
x² + (y+16)² = 39²
Subtract these equations to obtain an equation in y yields:
(y+16)² - y² = 39² - 25².
y² + 32y + 256 - y² = 896
32y = 640
y = 20
Plug this into the second equation to obtain:
x² + (20)² = 25²
x² = 225
x = 15

Now with these two numbers find the volume of the cone when A is revolved about CD (cone₁).
V(cone₁) = Bh/3
V(cone₁) = π(15)²(16+20)/3
V(cone₁) = 225π(36)/3
V(cone₁) = 2700π

Now find the volume of the cone when A is revolved about BD (cone₂):
V(cone₂) = Bh/3
V(cone₂) = π15²(20)/3
V(cone₂) = 225π(20)/3
V(cone₂) = 1500π

Subtracting the volume of cone₂ from cone₁ yielded the volume of A revolved about CB.
2700π - 1500π
1200π

Solution submitted by Roth