If the radius of a circular wave is increasing at 2 meters/sec, how fast is the area changing when the radius is 5 meters?
A)62.8 square meters per second
B)25.0 square meters per second
C)31.4 square meters per second
D)6.3 square meters per second
E) None of the above
To find how fast the area is changing, we need to use the formula for the area of a circle A = πr^2, where A is the area and r is the radius.
Given that the radius is increasing at a rate of 2 meters/sec, we can differentiate both sides of the equation with respect to time to find how the area is changing with respect to time.
dA/dt = d/dt(πr^2)
Using the chain rule, the derivative of r^2 with respect to time is 2r(dr/dt), where dr/dt is the rate at which the radius is changing. Plugging this into the equation, we get:
dA/dt = d/dt(πr^2) = 2πr(dr/dt)
Now we can substitute the given values. When the radius is 5 meters, we have r = 5 and dr/dt = 2 meters/sec. Plugging these values into the equation, we get:
dA/dt = 2π(5)(2) = 20π
To find the value in square meters per second, we can approximate π as 3.14.
dA/dt ≈ 20(3.14) ≈ 62.8 square meters per second
Therefore, the correct option is A) 62.8 square meters per second.