Question

A population of deer experienced 8.2% monthly growth compounded continuously for m months. At the end of the m days, the population had doubled in size.  The doubling-time for this population of deer is m months. Solve for m rounding to the nearest whole month.

Answer 1

The doubling time for this population of deer is approximately 8 months.

How to solve for the value of m

To solve for the value of m, use the formula for continuous compound interest:

`A = P * e^(r * t)`

In this case,

A represents the final population size,

P represents the initial population size,

r represents the growth rate, and

t represents the time in months.

Given that the population experienced a monthly growth rate of 8.2% (or 0.082 in decimal form), set up the equation as follows:

`2P = P * e^(0.082 * m)`

Here, we are doubling the initial population size (2P) and setting it equal to P multiplied by the continuous growth formula.

We can simplify the equation by canceling out the P on both sides:

`2 = e^(0.082 * m)`

To solve for m, isolate the exponent term.

Taking the natural logarithm (ln) of both sides of the equation will help us achieve that:

ln(2) = 0.082 * m

Now, solve for m by dividing ln(2) by 0.082:

m = ln(2) / 0.082

m ≈ 8.44

Rounding to the nearest whole month, the doubling time for this population of deer is approximately 8 months.