Volume Of A Sphere

Let the side of the cube be , and the radius of the sphere be . Given that the side of the cube is equal to the diameter of the sphere, we have: Now, we are asked to find the ratio of the increase in their surface areas. 1. Surface area of the cube: The surface area of a cube with side length is given by: 2. Surface area of the sphere: The surface area of a sphere with radius is given by: Next, we calculate the rate of increase in surface area. Let represent the rate of change of the side of the cube, and represent the rate of change of the radius of the sphere. Since the side and the radius are increasing at the same rate, we have: To find the rate of increase of the surface areas, we differentiate both surface area formulas with respect to time. 1. Rate of change of surface area of the cube: 2. Rate of change of surface area of the sphere: Now, we compute the ratio of the rate of increase of the surface areas: Since , we substitute into the equation: Thus, the ratio of the increase in their surface areas is . Hence, the correct answer is (A).

Let the radius of the sphere be . The surface area and the volume of a sphere are given by the formulas: Now, we want to find the rate of change of the volume with respect to the surface area. First, we differentiate both expressions with respect to : The rate of change of the volume with respect to the surface area is: Now, using the formula for the surface area , we solve for : Substitute this value of into the expression for : Thus, the correct answer is (D).