The rate at which people enter an amusement park on a given day is modeled by the function E defined by:
E(t) = 15600/(t² - 24t + 160).
The rate at which people leave the same amusement park on the same day is modeled by the function L defined by:
L(t) = 9890/(t² - 38t +370).
Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 9 < t < 23, the hours during which the park is open. At time t = 9, there are no people in the park.
How many people have entered the park by 5:00 pm (t=17)? Round your answer to the nearest whole number.
The Attempt at a Solution
This problem is actually quite simple. I know I need to take the integral of that function and then solve for C using the information they give me, then plug 17 into the integrated equation.
The only problem is.. integrating that equation is difficult.
I know I can factor out the 15600 in E(t) to get
15600 x ln |(t² - 38t +370)| (because the anti-derivative of 1/x is ln |x|)
That is where I am stuck right now. I know that that isn't the full, complete answer, because if I take the derivative of that equation to check, it doesn't match, because of the chain rule. Am I going about this the wrong way? I don't see any other method, because u-substitution doesn't work, long division won't help, and I don't see how I can break down that equation any further.
I would REALLY like to avoid guess-and-check unless that is the only absolute way I can solve this (which I doubt).
Thanks to any help.. =)
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