Oil flows into a tank according to the rate F of t equals the quotient of t squared plus 1 and the quantity 1 plus t , and at the same time empties out at the rate E of t equals the quotient of the...

Question

Oil flows into a tank according to the rate  , and at the same time empties out at the rate  , with both F(t) and E(t) measured in gallons per minute. How much oil, to the nearest gallon, is in the tank at time t = 12 minutes? You must show your setup, but can use your calculator for all evaluations.

Answer 1

This looks fairly straightforward.

F(t) = (t² + 1)/(1 + t) gal/min

E(t) = (ln(t + 7))/(t + 2) gal/min

At t = 12 min

F(12) = ((12)² + 1)/(12 + 1) = (144 + 1)/(13) = 145/13 gal ≈ 11.15 gals

So after 12 minutes, 11.15 gallons enters the tank.

E(12) = ln(12 + 7)/(12 + 2) = [ln(19)]/(14) ≈ 0.21 gals.

After 12 minutes, 0.21 gallons has emptied out of the tank.

F(12) - E(12) = 11.15 - 0.21 = 10.94 ≈ 11 gallons is in the tank after 12 minutes.