The radius of a circle is decreasing at a constant rate of 0.1 centimeters per second. In terms of the circumference C, what is the rate of the area of the circle, in square centimeters per second?


a. -(0.2)pie C
b. -(0.1)C
c. -(0.1)C/2 pie
d. (0.1)^2C
e. (0.1)^2 pie C

See:

http://www.jiskha.com/display.cgi?id=1295396006

To find the rate of change of the area of a circle, we need to use the formulas for the circumference and the area of a circle.

The formula for the circumference of a circle is C = 2πr, where C represents the circumference and r represents the radius.

The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius.

Given that the radius is decreasing at a constant rate of 0.1 centimeters per second, we can represent this rate as dr/dt = -0.1 cm/s, where dr/dt represents the rate of change of the radius with respect to time.

We need to find the rate of change of the area, dA/dt, in terms of the circumference C.

We can use the chain rule of differentiation to relate the rate of change of the area to the rate of change of the radius.

The chain rule states that if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.

In this case, we have A = πr^2, so u = r and y = A.

Differentiating both sides with respect to time t, we get dA/dt = dA/dr * dr/dt.

Now, we need to express dA/dr in terms of C.

From the equation A = πr^2, we can find dr/dA by differentiating both sides with respect to r:

2πr * dr/dr = dA/dr
2πr = dA/dr
dA = 2πr * dr

Substituting -0.1 for dr/dt, we get:

dA/dt = 2πr * (-0.1)

Since C = 2πr, we can substitute it into the equation:

dA/dt = -0.1 * C

Therefore, the rate of change of the area of the circle, in square centimeters per second, is -0.1C.

So, the correct answer is b. -(0.1)C.