The number of gallons of water remaining in a swimming pool t minutes after the drains are opened and the pool begins emptying is give given by the function f(x)=40t^2-3000t+40,000gallons. Exactly 2 minutes after the drains are opened, the pool was emptying at an instantaneous rate of _____ gallons per minute. Find the instantaneous rate of change at t =2.
To find the instantaneous rate of change at t = 2, we need to take the derivative of the function f(t) with respect to t and evaluate it at t = 2.
First, let's find the derivative of the function f(t) = 40t^2 - 3000t + 40,000.
To do this, we can use the power rule for derivatives. According to the power rule, if we have a term of the form ax^n (where a is a constant and n is any real number), the derivative of this term with respect to x is given by nx^(n-1).
Applying the power rule to each term in f(t):
The derivative of the first term 40t^2 is 80t^(2-1) = 80t.
The derivative of the second term -3000t is -3000.
The derivative of the third term 40,000 is 0 (since it's a constant).
So, the derivative of the function f(t) = 40t^2 - 3000t + 40,000 is f'(t) = 80t - 3000.
Now, let's evaluate the derivative at t = 2:
f'(2) = 80(2) - 3000
= 160 - 3000
= -2840 gallons per minute.
Therefore, the instantaneous rate of change at t = 2 is -2840 gallons per minute.