Question

WORTH 30 POINTS

A pool drains at a rate of 3% every 5 minutes. If it starts with 1500 gallons, how many gallons are left in the pool after 50 minutes? About how many minutes will it take to have less than 1000 gallons in the pool? Justify your answer with a mathematical model of the problem situation.

Answer 1

Using an exponential function, it is found that:

• 1,106 gallons of water will be in the pool after 50 minutes.

• It will take 66.56 minutes for there to be less than 1000 gallons on the pool.

What is an exponential function?

A decaying exponential function is modeled by:

$A(t) = A(0)(1 - r)^t$

In which:

• A(0) is the initial value.

• r is the decay rate, as a decimal.

Considering that the pool drains at a rate of 3% every 5 minutes, and starts with 1500 gallons, the amount of water in the pool after t minutes is given by:

$A(t) = 1500(0.97)^{0.2t}$

The multiplication by 0.2 is because 1/5 = 0.2.

The amount after 50 minutes is given by:

$A(50) = 1500(0.97)^{0.2(50)} = 1106$

1,106 gallons of water will be in the pool after 50 minutes.

To find when the pool will have less than 1000 gallons, we solve for t when A(t) = 1000, hence:

$A(t) = 1500(0.97)^{0.2t}$

$1000 = 1500(0.97)^{0.2t}$

$(0.97)^{0.2t} = \frac{2}{3}$

$\log{(0.97)^{0.2t}} = \log{\left(\frac{2}{3}\right)}$

$0.2t\log{(0.97)} = \log{\left(\frac{2}{3}\right)}$

$t = \frac{\log{\left(\frac{2}{3}\right)}}{0.2\log{(0.97)}}$

t = 66.56.

It will take 66.56 minutes for there to be less than 1000 gallons on the pool.

More can be learned about exponential functions at https://brainly.com/question/25537936

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