The radius of a circular oil spill is increasing at a rate of 10 feet per hour. How quickly is the surface area changing when the radius is 200 feet?

area= PI r^2

darea/dt=PI 2 r dr/dt

solve for d(area)/dt. You know r, and dr/dt

To find how quickly the surface area is changing with respect to time, we can differentiate the formula for the area of a circle with respect to the radius.

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

Differentiating both sides of the equation with respect to time (t), we get:

dA/dt = d/dt (πr^2)

Now, let's find the value of dA/dt when the radius (r) is 200 feet.

To do this, we need to substitute r = 200 into the equation dA/dt = d/dt (πr^2).

dA/dt = d/dt (π(200)^2)

= d/dt (π(40000))

= π * d/dt (40000)

Since the radius is increasing at a constant rate of 10 feet per hour, its derivative with respect to time is 10. Hence, d/dt (40000) = 10.

dA/dt = π * 10

= 10π

So, when the radius is 200 feet, the surface area of the oil spill is changing at a rate of 10π square feet per hour.

To find the rate at which the surface area is changing with respect to time, we can use the formula for the surface area of a circle:

Surface Area = π * r^2

Where r is the radius of the circle.

We are given that the rate at which the radius is increasing is 10 feet per hour, which means that dr/dt = 10 ft/h.

To find how quickly the surface area is changing when the radius is 200 feet, we need to find dA/dt at r = 200 feet.

Since A = π * r^2, we can find dA/dt by finding the derivative of A with respect to t:

dA/dt = dA/dr * dr/dt

To find dA/dr, we differentiate the formula for the area of a circle with respect to r:

dA/dr = 2 * π * r

Now we can substitute our known values into the formula:

dA/dt = (2 * π * r) * (dr/dt)

Plugging in r = 200 ft and dr/dt = 10 ft/h:

dA/dt = (2 * π * 200) * 10

Simplifying gives us:

dA/dt = 4000π ft^2/h

Therefore, the surface area is changing at a rate of 4000π square feet per hour when the radius is 200 feet.