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
a
To find the rate of change of the area of a circle, we need to find the derivative of the area with respect to time.
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
Given that the radius is decreasing at a constant rate of 0.1 centimeters per second, we can write the relationship between the radius and time as r = r0 - 0.1t, where r0 is the initial radius and t is the time.
We can substitute this relationship into the area formula to get A = π(r0 - 0.1t)^2.
To find the rate of change of the area, we differentiate the area formula with respect to time:
dA/dt = d/dt [π(r0 - 0.1t)^2]
Using the chain rule, we get:
dA/dt = 2π(r0 - 0.1t)(-0.1)
Simplifying the expression, we get:
dA/dt = -0.2π(r0 - 0.1t)
Now, we need to express the result in terms of the circumference C. The circumference of a circle is given by the formula C = 2πr.
Since r0 is the initial radius, we can express it in terms of C as r0 = C/(2π).
Substituting this relationship into the expression for dA/dt, we get:
dA/dt = -0.2π(C/(2π) - 0.1t)
Simplifying further, we have:
dA/dt = -0.1(C - πt)
Therefore, the rate of change of the area of the circle in terms of the circumference C is given by option b. -(0.1)C.