The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $= \sqrt{3}$ is :
- $\displaystyle \frac{4}{3} \sqrt{3} \pi$
- $2 \pi$
- $\displaystyle \frac{8}{3} \sqrt{3} \pi$
- $4 \pi$
For a cylinder of radius r and height h inscribed in a sphere of radius R, r^2 + (h/2)^2 = R^2. Volume V = pi * r^2 * h = pi * (R^2 - h^2/4) * h = pi * (R^2*h - h^3/4). Maximize V by setting dV/dh = 0: R^2 - 3h^2/4 = 0, so h^2 = 4R^2/3, h = 2R/sqrt(3). Then r^2 = R^2 - R^2/3 = 2R^2/3. Max V = pi * (2R^2/3) * (2R/sqrt(3)) = 4*pi*R^3 / (3*sqrt(3)). With R=sqrt(3), V = 4*pi*(3*sqrt(3)) / (3*sqrt(3)) = 4*pi.
For the largest right circular cylinder inscribed in a sphere of radius R, the height h of the cylinder must equal 2R divided by the square root of 3. Given R = square root of 3, the height h is 2. The radius r of the cylinder's base is 1, calculated from the relation r squared equals R squared minus (h/2) squared. Substituting r = 1 and h = 2 into the cylinder volume formula, V = pi * r squared * h, gives a volume of 4 * pi.