Question

a beach ball is deflating at a constant rate of 10 cubic centimeters per second. when the volume of the ball is 256/3pie cubic centimeters, what is the rate of change of the surface area

Answer 1

To find the rate of change of the surface area, first find the radius of the beach ball by finding the cube root of its volume. Then, use the formula for the surface area of a sphere to calculate the rate of change of the surface area. The rate of change of the surface area is approximately 462.35 square centimeters per second.

Step-by-step explanation:

To find the rate of change of the surface area, we need to first find the radius of the beach ball. We can do this by finding the cube root of the volume of the ball. Given that the volume of the ball is 256/(3π) cubic centimeters, the radius can be found as follows:

Let V be the volume of the ball.

V = 256/(3π)

V = 256/(3*3.14)

V ≈ 27.33

The cube root of V is approximately 3.42 centimeters.

Next, we can find the rate of change of the surface area. The surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere.

So, the rate of change of the surface area is 4π(3.42)^2 multiplied by the rate at which the ball is deflating, which is 10 cubic centimeters per second.

The rate of change of the surface area is approximately 462.35 square centimeters per second.

Answer 2

To determine the rate of change of the surface area of a deflating beach ball, we use calculus to relate the volume and surface area of the sphere and apply the given rate of volume loss.

Step-by-step explanation:

To find the rate of change of the surface area of the beach ball as it deflates, we need to establish the relationship between the volume and the surface area of a sphere. The formula for the volume of a sphere is V = (4/3)πr^3 and the formula for the surface area is A = 4πr^2. Differentiating both equations with respect to time t gives us dV/dt and dA/dt, which represent the rate of change of volume and surface area, respectively.

We are given that dV/dt = -10 cm^3/s. To find dA/dt, we first need to express r in terms of V and then differentiate A with respect to t using the chain rule. Since we have the volume of the ball at a certain instant, we can find r and then find dA/dt at the precise moment when V is 256/3π cm^3.