Question

The radius of a circle is decreasing at a

constant rate of 7 centimeters per
second. At the instant when the radius
of the circle is 4 centimeters, what is the
rate of change of the area? Round your
answer to three decimal places​

Answer 1

To find the rate of change of the area of a circle, differentiate the formula for the area with respect to time. The rate of change is equal to -14πr cm^2/s.

Step-by-step explanation:

To find the rate of change of the area of a circle, we need to differentiate the formula for the area of a circle with respect to time. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.

Using the chain rule, we can differentiate both sides of the equation. The derivative of A with respect to time (dA/dt) is equal to (dA/dr) * (dr/dt). Since the radius is decreasing at a constant rate of 7 centimeters per second, we can substitute dr/dt with -7 cm/s.

Therefore, the rate of change of the area of the circle is given by: dA/dt = 2πr * (-7) = -14πr cm^2/s.

Answer 2

Let's start by recalling the formula for the area of a circle: A = πr^2, where A is the area and r is the radius.

We are given that the radius is decreasing at a constant rate of 7 cm/s. This means that dr/dt = -7 cm/s, where t is time.

We want to find the rate of change of the area, dA/dt, at the instant when r = 4 cm.

To do this, we can use the chain rule of differentiation:

dA/dt = dA/dr * dr/dt

We know that A = πr^2, so we can differentiate with respect to r to find dA/dr:

dA/dr = 2πr

Now we can substitute the given values and solve for dA/dt:

r = 4 cm

dr/dt = -7 cm/s

dA/dr = 2πr = 2π(4) = 8π

dA/dt = dA/dr * dr/dt = (8π) * (-7) = -56π ≈ -175.929 cm^2/s (rounded to three decimal places)

Therefore, the rate of change of the area is approximately -175.929 cm^2/s when the radius of the circle is 4 cm and decreasing at a constant rate of 7 cm/s.