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?
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To find out how quickly the surface area of the oil spill is changing, we can use the formula for the surface area of a circle:
A = π * r^2
Where A represents the surface area and r represents the radius.
To find the rate at which the surface area is changing, we need to take the derivative with respect to time of the surface area equation:
dA/dt = d(π * r^2)/dt
To do this, we need to apply the chain rule. The chain rule states that if we have a composite function, such as f(g(x)), then the derivative of the composite function is the derivative of the outer function multiplied by the derivative of the inner function. In this case, our composite function is A(r(t)), where the radius is a function of time.
So, applying the chain rule to our equation, we have:
dA/dt = dA/dr * dr/dt
Now, let's find the derivatives:
First, let's find the derivative of the surface area with respect to the radius, dA/dr. Applying the power rule, we have:
dA/dr = 2πr
Next, let's find the derivative of the radius with respect to time, dr/dt. We are given that the radius is increasing at a rate of 10 feet per hour. So:
dr/dt = 10 ft/hr
Now we can substitute these values into our chain rule equation:
dA/dt = (2πr) * (dr/dt)
Since we are asked to find the rate of change of the surface area when the radius is 200 feet, we can substitute r = 200 into the equation:
dA/dt = (2π * 200) * (10 ft/hr)
Now we can calculate it:
dA/dt = 4000π ft²/hr
So, when the radius is 200 feet, the surface area is changing at a rate of 4000π square feet per hour.