Aswimming pool is nearly empty, holding only 5400 gallons of water. a system is set up so that the water in the pool starts to increase by 13% per hour. after how many hours is the pool filling at a rate of 2848 gallons per hour?

  11.964 hours

Step-by-step explanation:

The number of gallons p in the pool after x hours can be written as ...

  p(x) = 5400·1.13^x

The derivative of this is ...

  p'(x) = 5400·1.13^x·ln(1.13)

We want that equal to 2848, so we want to solve ...

  2848 = 5400·1.13^x·ln(1.13)

Dividing by the coefficient of the exponential term gives ...

  2848/(5400·ln(1.13)) = 1.13^x

Taking the natural log, we get ...

  ln(2848/(5400·ln(1.13))) = x·ln(1.13)

  x = ln(2848/(5400·ln(1.13))) / ln(1.13) ≈ 11.9637 . . . hours

After about 11.964 hours, the pool is filling at the rate 2848 gph.