Question

When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.02c(m)/(m)in. At what rate is the plate's area increasing when the radius is 56cm ?

Answer 1

The rate at which the plate's area is increasing when the radius is 56 cm is approximately 7.04π cm²/min.

Step-by-step explanation:

To find the rate at which the plate's area is increasing when the radius is 56 cm, we can use the formula for the area of a circle: A = πr², where A represents the area and r represents the radius. We are given the rate at which the radius is increasing, which is 0.02 cm per minute. Using the chain rule, we can differentiate both sides of the equation to find the rate at which the area is changing with respect to time.

Let's differentiate both sides of the equation with respect to time: dA/dt = 2πr(dr/dt). Substitute the given values: dA/dt = 2π(56)(0.02). Simplify: dA/dt = 2π(1.12) cm²/min. Therefore, the plate's area is increasing at a rate of approximately 7.04π cm²/min when the radius is 56 cm.