Question

Water is poured into a bucket according to the rate F(t)=(t+7)/(t+2) , and at the same time empties out through a hole in the bottom at the rate E(t)=(ln(t+4))/t+2 , with both F(t) and E(t) measured in pints per minute. How much water, to the nearest pint, is in the bucket at time t = 5 minutes. You must show your setup but can use your calculator for all evaluations.

Answer 1

I have no idea how to do this problem the only thing I know is that it will equal the definite integral from [0,5] of F(t)-E(t)dt

Answer 2

correct. So, just do it...

∫[0,5] F(t)-E(t) dt
= ∫[0,5] (t+7)/(t+2) - ln(t+4)/(t+2) dt

∫(t+7)/(t+2) dt = ∫1 + 5/(t+2) dt
= t + 5ln(t+2)

For ∫ln(t+4)/(t+2) dt, you're in trouble. It is not doable with elementary functions. You sure there's no typo?

∫ln(u)/u du = 1/2 ln^2(u)

but you don't quite have that.

Answer 3

I can ask my teacher! how would we do it if it was actually just (t-4)/(t+2)?

Answer 4

oops I meant (t+4)/(t+2)

Answer 5

then it's just like the one I showed first. divide and hen integrate.