Question

Use the approximate half-life formula for the case described below. Discuss whether the formula is valid for the case described.Urban encroachment is causing the area of a forest to decline at the rate of 9% per year. What is the half-life of the forest? What fraction of the forest will remain in 30 years?(Type an integer or decimal rounded to the nearest hundredth as needed.)

Answer 1

Answer:

Half-life = 7.35 years

After 30 years 0.06 of the forest will remain

Explanation:

Half-life is the amount of time it takes the forest to decline to half its initial value.

Now we are told that the forest declines at a rate of 9% per year. This means the amount left next year is 100% - 9% = 91% of the previous. Therefore, if we call the initial amount A, then the amount left after t years will be

$P(t)=A(\frac{91\%}{100\%})^t$$\Rightarrow P(t)=A(0.91)^t$

Now, when the forest declines to half its initial value, we have

$\frac{A}{2}=A(0.91)^t$

Canceling A from both sides gives

$\frac{1}{2}=0.91^t$

Taking the logarithm (of base 0.91) of both sides gives

$\log_{0.91}(\frac{1}{2})=t$$t=7.35\text{ years.}$