Question

Water is being pumped continuously from a pool at a rate proportional to the amount of water left in the pool. Initially there was 15,000 gallons of water in the pool; six minutes later there was 13,800 gallons.

At what rate was the amount of water in the pool decreasing when there were 14,000 gallons remaining and when will there be 5,000 gallons remaining?

Please show all steps.

Answer 1

• 194.6 gpm at 14,000 gallons
• 69.5 gpm at 5,000 gallons.

Explanation:

When a value is decreasing at a rate proportional to that value, it can be modeled by the formula

a = a0·e^(-kt)

where k is the constant of proportionality.

Alternatively, we can write the exponential function describing the pool volume* as ...

a = 15000·(138/150)^(t/6) = 15000·((138/150)^(1/6))^t

Comparing these, we see that ...

e^(-kt) = (138/150)^(t/6)

or ...

k = -ln(138/150)/6 ≈ 0.0138969

__

So, when 14000 gallons remain, the rate of decrease is ...

14000·0.0138969 ≈ 194.6 . . . gallons per minute

When 5000 gallons remain, the rate of decrease is ...

5000·0.0138969 ≈ 69.5 . . . gallons per minute

_____

* The generic form of this is ...

(initial value) · (multiplier per interval)^(number of intervals)

Here, the multiplier over a 6-minute period is 13800/15000 = 138/150, and the number of 6-minute intervals is t/6 when t is in minutes.

_____

Effectively, we make use of the fact that for ...

a = a0·e^(-kt)

the derivative is ...

da/dt = -k(a0·e^(-kt)) = -k·a

That is, k is the constant of proportionality mentioned in the first sentence of the problem statement.