A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?

Verified step by step guidance

1. First, recall the formula for the area of a circle: A = πr², where A is the area and r is the radius.
2. To find the rate of change of the area with respect to time, we need to differentiate the area formula with respect to time. This involves using the chain rule since the radius is changing with time.
3. Apply the chain rule: dA/dt = dA/dr * dr/dt. Here, dA/dr is the derivative of the area with respect to the radius, and dr/dt is the rate at which the radius is changing.
4. Calculate dA/dr: Differentiate A = πr² with respect to r to get dA/dr = 2πr.
5. Substitute the values: Use r = 10 ft and dr/dt = -2 ft/min (since the radius is decreasing) into the equation dA/dt = 2πr * dr/dt to find the rate of change of the area at the instant the radius is 10 ft.

Key Concepts

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the area of the circle changes as the radius decreases. This requires applying the chain rule from calculus to relate the rates of change of the radius and the area.

Area of a Circle

The area of a circle is calculated using the formula A = πr², where A is the area and r is the radius. Understanding this formula is crucial because we need to differentiate it with respect to time to find how the area changes as the radius changes over time.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. In this context, we will differentiate the area formula with respect to time to find the rate of change of the area as the radius changes, applying the chain rule to connect the rates of change.