Question

in a country, deforestation occurs at the rate of about 3.6% per year. assume that the amount of forest remaining is determined by the function $f = f_0e^{-0.036t}$, where $f_0$ is the present acreage of forest land and $t$ is the time in years from the present. in how many years will there be only 20% of the present acreage remaining? hint: find the amount of time it takes for $f_0$ to become $0.20f_0$.

there will be only 20% of the present acreage remaining in \\(\square\\) years.
(type an integer or decimal rounded to one decimal place as needed.)

Explanation:

Step1: Set up the equation

We know that \( F = F_0e^{-0.036t} \) and we want to find \( t \) when \( F = 0.2F_0 \). So substitute \( F = 0.2F_0 \) into the equation:
\( 0.2F_0 = F_0e^{-0.036t} \)

Step2: Divide both sides by \( F_0 \)

Since \( F_0
eq 0 \) (it's the present acreage of forest land), we can divide both sides by \( F_0 \):
\( 0.2 = e^{-0.036t} \)

Step3: Take the natural logarithm of both sides

Take \( \ln \) of both sides:
\( \ln(0.2) = \ln(e^{-0.036t}) \)
Using the property \( \ln(e^x)=x \), the right side becomes \( -0.036t \):
\( \ln(0.2) = -0.036t \)

Step4: Solve for \( t \)

Divide both sides by \( -0.036 \):
\( t=\frac{\ln(0.2)}{-0.036} \)
Calculate \( \ln(0.2)\approx - 1.6094 \), then:
\( t=\frac{-1.6094}{-0.036}\approx44.7 \)

Answer:

44.7