Oil from a ruptured tanker spreads in a circular pattern. If the radius of the circle increases at the constant rate of 1.5 feet per second, how fast is the enclosed area increasing at the end of 2 hours?
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To find how fast the enclosed area is increasing, we need to use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.
We are given that the radius of the circle is increasing at a constant rate of 1.5 feet per second. This means that dr/dt = 1.5 feet per second.
We need to find how fast the enclosed area is increasing, which is dA/dt. To do this, we differentiate the area formula with respect to time:
dA/dt = d/dt(πr^2)
To find dA/dt, we need to use the chain rule. The chain rule states that if we have a function of a function, then the derivative is given by the derivative of the outer function multiplied by the derivative of the inner function.
Applying the chain rule to our problem, we have:
dA/dt = dA/dr * dr/dt
We know that dA/dr is simply the derivative of the area formula, which is:
dA/dr = d/dr(πr^2) = 2πr
And we already know that dr/dt is 1.5 feet per second.
Now we can substitute these values into the equation:
dA/dt = (2πr) * (1.5)
We are asked to find how fast the enclosed area is increasing at the end of 2 hours. Since we are given the rate of increase in terms of feet per second, we need to convert 2 hours into seconds.
There are 60 minutes in an hour and 60 seconds in a minute, so 2 hours is equal to 2 * 60 * 60 = 7200 seconds.
Now we can substitute the values into the equation:
dA/dt = (2πr) * (1.5)
At the end of 2 hours, we need to know the value of r to find the rate of increase. Unfortunately, we are not given the initial radius. Without knowing the starting point, we cannot determine the specific value of dA/dt.