Question

Water is poured into a bucket according to the rate F of t equals the quotient of 7 plus t and the quantity 2 plus t, and at the same time empties out through a hole in the bottom at the rate E of t equals the quotient of the natural log of the quantity t plus 4 and the quantity t plus 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

https://gyazo.com/d2a5ac63b91f596e6febfa3e8dbd91aa

Answer 2

@jim_thompson5910 @SolomonZelman

Answer 3

\(\large\color{#000000 }{ \displaystyle F(t)=\frac{t+7}{t+2} }\)

\(\large\color{#000000 }{ \displaystyle E(t)=\frac{\ln(x+4)}{t+2} }\)

Answer 4

are the functions like this?

Answer 5

Yes

Answer 6

The amount of water is the area under the curve....

\(\large\color{#000000 }{ \displaystyle \int \left[F(t)-E(t)\right]dt }\)

Answer 7

Ohh. Ok. I got it from here

Answer 8

Unless you feel like doing it for me, I don't mind :p

Answer 9

EXAMPLE OF THE SAME CONCEPT:

Same way if you were given the velocities of the cars,
\(y_1(x)\) for car 1 and \(y_2(x)\) for car 2, then you
will need an integral of \(y_1(x)-y_2(x)\) to find how
much distance is car 1 ahead of car 2.

(provided that car 1 is indeed ahead)

Answer 10

I do like integrating, especially if someone approaches and tells me that the problem is impossible. Even more do I enjoy using elemnatry properties for harder problems, but according to the policy I can't do it. (Afterall you have to integrate too)

Answer 11

oh, forgot to mention your limits of integration are 0 and 5.

Answer 12

because you are looking for the water in the tank in 5 minutes.

Answer 13

Ughh I have to use quotient rule right?

Answer 14

\(\large\color{#000000 }{ \displaystyle \int_0^5\left[\frac{t+7}{t+2} -\frac{\ln(x+4)}{t+2} \right]dt }\)

Answer 15

Oh yeah I have a neato calculator no need for me to use quotient rule.

Answer 16

9.056 final answer

Answer 17

you can't use the quotient rule

Answer 18

it is integration (not differentiation)

Answer 19

\(\large\color{#000000 }{ \displaystyle \int_0^5\left[\frac{t+7}{t+2} -\frac{\ln(t+4)}{t+2} \right]dt }\)

Answer 20

Oh my mistake. If I wanted to manually do it I would have subtract the two and find a way to simplify it so I may integrate.

Answer 21

I hope you're not integrating (unless you did it for fun) as I am aware of how it is done, no need to teach me :)

Answer 22

I don't you think you can hande the last component ln(t+4)/(t+2).
If it were ln(t+4)/(t+4) or ln(t+2)/(t+2), then you would.

Answer 23

are you familiar with polylogarithmic functions?

Answer 24

No, I am not. I don't think I need to be either at least not in my current course.