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+40,000 gallons. Between 3 and 5 minutes after the drains were opened, the pool emptied at an average rate of ___ gallons per minute. Find the average rate of change between t=3 and t=5
To find the average rate of change between t = 3 and t = 5, we need to calculate the change in the function f(x) divided by the change in time.
The change in the function f(x) between t = 3 and t = 5 is f(5) - f(3). Let's calculate this:
f(5) = 40(5)^2 - 3000(5) + 40,000 = 40(25) - 15,000 + 40,000 = 1,000 - 15,000 + 40,000 = 26,000 gallons
f(3) = 40(3)^2 - 3000(3) + 40,000 = 40(9) - 9,000 + 40,000 = 360 - 9,000 + 40,000 = 31,360 gallons
Therefore, the change in the function f(x) between t = 3 and t = 5 is 26,000 - 31,360 = -5,360 gallons.
The change in time between t = 3 and t = 5 is 5 - 3 = 2 minutes.
The average rate of change is obtained by dividing the change in the function by the change in time:
Average rate of change = (Change in f(x)) / (Change in time) = (-5,360 gallons) / (2 minutes)
Dividing, we find that the average rate of change between t = 3 and t = 5 is -2,680 gallons per minute.