# A right circular cylinder is inscribed in a cone with height 6m and radius 3m. How do you find the largest possible volume of such a cylinder?

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To find the largest possible volume of the inscribed cylinder, maximize the volume function V(r) = πr^2h, where r is the radius of the cylinder and h is the height. The radius of the cylinder can be expressed in terms of the height of the cone and its radius using similar triangles. Thus, the maximum volume occurs when the cylinder is tangent to the cone's curved surface. Solving for the radius of the cylinder gives r = (3/2)h. Substituting this expression for r into the volume function, we get V(h) = (27/4)πh^3. To maximize V(h), find its critical points by taking the derivative, setting it equal to zero, and solving for h. Differentiate V(h) with respect to h and set it equal to zero: dV/dh = (81/4)πh^2 = 0. Solve for h to find the critical point: h = 0. Then, use the second derivative test to confirm that this critical point is a maximum. Since the second derivative is positive, the critical point is a maximum. Thus, the largest possible volume of the inscribed cylinder is V(6) = (27/4)π(6)^3 = 324π cubic meters.

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