Question

A forest fire spreads in a circular pattern with the radius changing at the rate of 50 miles per hour. At what rate is the fire spreading when the diameter of the fire is 310 miles?

A) 48.694 m²/h.
B) 155 m²/h.
C) 15,500 m²/h.
D) 48,694.69 m²/h.

Answer 1

The forest fire is spreading at a rate of 48,694.69 square miles per hour when the diameter is 310 miles. To calculate this, we use the area formula of a circle and the chain rule for derivatives.

Step-by-step explanation:

The question asks about the rate at which a forest fire is spreading when its diameter is 310 miles. Since the forest fire spreads in a circular pattern, we need to use the formula for the area of a circle A = πr^2, where r is the radius of the circle. Given the rate at which the radius is changing (50 miles per hour), we can find the rate at which the area is increasing, also known as the derivative of A with respect to time (dA/dt).

First, we must convert the diameter to radius by dividing it by 2, giving us a radius (r) of 155 miles for the fire. Then, to find dA/dt, we use the chain rule, which in this case is dA/dt = 2πr(dr/dt). Plugging in our values, we get dA/dt = 2π(155 miles)(50 miles/hour).

After calculating this expression, we find the correct answer to be 48,694.69 square miles per hour, which corresponds to option D.