Question

The initial population of a town is 2600, and it grows with a doubling time of 10 years. Determine how long it will take for the population to quadruple.
It will take the population years to quadruple.
(Round to the nearest whole number as needed.)
Enter your answer in the answer box.
ABD

Answer 1

Answer:

It will take the 20 years for the population to quadruple.

Step-by-step explanation:

Exponential population growth:

An exponential model for population growth has the following model:

$P(t) = P(0)(1+r)^t$

In which P(0) is the initial population and r is the growth rate, as a decimal.

It grows with a doubling time of 10 years.

This means that $P(10) = 2P(0)$. We use this to find r. So

$P(t) = P(0)(1+r)^t$

$2P(0) = P(0)(1+r)^10$

$(1+r)^10 = 2$

$\sqrt[10]{(1+r)^10} = \sqrt[10]{2}$

$1+r = 2^{\frac{1}{10}}$

$1 + r = 1.0718$

So

$P(t) = P(0)(1+r)^t$

$P(t) = P(0)(1.0718)^t$

Determine how long it will take for the population to quadruple.

This is t for which P(t) = 4P(0). So

$P(t) = P(0)(1.0718)^t$

$4P(0) = P(0)(1.0718)^t$

$(1.0718)^t = 4$

$\log{(1.0718)^t} = \log{4}$

$t\log{1.0718} = \log{4}$

$t = \frac{\log{4}}{\log{1.0718}}$

$t = 20$

It will take the 20 years for the population to quadruple.