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 natural log of the quantity t plus 7 and the quantity t plus 2, 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: 10.94 gallons

Step-by-step explanation:

The rate of flow is given by the equation:

F (t) = (t^2+1) / (t + 1)

The rate at which it empties:

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

Where t represr ts time in both equations

Amount of oil, to the nearest gallon, is in the tank at time t = 12 minutes

We can get this by taking the difference of both equations since it is occurring at the same time 't'

That is : F (t) - E (t)

At t = 12

[ (t^2+1) / (t + 1) ] - [ ln (t + 7) / (t + 2) ]

F (12) = (12^2 + 1) / (12 + 1) = 145/13 = 11.1538

E (12) = In (12 + 7) / (12 + 2) = In (19) / 14 =

E (12) = 2.9444389 / 14 = 0.2103

F (t) - E (t) = (11.1538 - 0.2103) = 10.9435 gallons