Question: Water is poured into a bucket according to the rate F(t)=(t+7)/(2+t), 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...

Question: Water is poured into a bucket according to the rate F(t)=(t+7)/(2+t), 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. 

How will I answer this question using integrals?

Asked on by sulbhe

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The rate at which water is poured into a bucket is given by the function `F(t)=(t+7)/(2+t)` , and at the rate at which the bucket empties out through a hole in the bottom is given by the function `E(t)=(ln(t+4))/(t+2)` , with both F(t) and E(t) measured in pints per minute.

The net increase in the amount of water in the bucket at time t is given by `F(t) - E(t)`

= `(t+7)/(2+t) - (ln(t+4))/(t+2)`

If the bucket was empty at time t = 0, the amount of water in the bucket at time t = 5 is given by the integral

`int_0^5 (t+7)/(2+t) - (ln(t+4))/(t+2) dt`

= `int_0^5 (t+7)/(2+t) dt - int (ln(t+4))/(t+2) dt`

The integral `int_0^5 (t+7)/(2+t) dt = [5*ln(t+2)+t + C]_0^5 `

and `int (ln(t+4))/(t+2) dt = [ln(t+4)*ln(1 - (t+4)/2) + Li_2((t+4)/2)+C]_0^5`

Here `Li_2(x)` refers to the polylogarithm function and can be expressed as a power series `Li_2 x = sum_(k=1)^oo (x^k/k^2)` .

The exact value of the integral to the required accuracy can be determined by evaluating the power series.

Sources:

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