Final answer:
The rate at which the area of the oil spill is changing can be found using the chain rule. Using the formula for the area of a circle, we can differentiate with respect to time to find the rate of change of the area over time. Plugging in the given values, we find that the rate of change is 5π m²/min.
Step-by-step explanation:
In this question, we are given that an engineer discovers an oil spill from a tank with a circular shape, and she measures the diameter of the spill as 10m. The diameter is growing at a constant rate of 0.5m/min. To determine the area of the spill over time, we can use the formula for the area of a circle: A = πr^2, where r is the radius of the spill. Since the diameter is given, we can find the radius by dividing the diameter by 2. Given that the radius is growing at a rate of 0.5m/min as the diameter grows, we can use the chain rule to find the rate of change of the area over tim
We start by finding the radius at any given time using the formula r = d/2, where d is the diameter. So, in this case, the radius at any given time (t) is r(t) = 10/2 = 5m.Next, we differentiate both sides of the equation A = πr^2 with respect to time (t) using the chain rule. The derivative of the area with respect to time gives us the rate at which the area of the spill is changing.
Using the chain rule, we have:
dA/dt = dA/dr * dr/dt
The first part, dA/dr, is found by differentiating the equation A = πr^2 with respect to r:
dA/dr = d(πr^2)/dr = 2πr
The second part, dr/dt, is the rate at which the radius is changing over time. In this case, we are given that the radius is growing at a constant rate of 0.5m/min. So, dr/dt = 0.5m/min.
Now we can substitute the values we obtained back into the chain rule equation:
dA/dt = 2πr * (dr/dt)
Plugging in the values of r and dr/dt, we have:
dA/dt = 2π * 5 * 0.5 = 5π m²/min
Therefore, the rate at which the area of the oil spill is changing is 5π m²/min.