Question

in a country, deforestation occurs at the rate of about 5.2% per year. assume that the amount of forest remaining is determined by the function f = f₀e⁻⁰·⁰⁵²ᵗ, where f₀ 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 60% of the present acreage remaining? hint: find the amount of time it takes for f₀ to become 0.60f₀. there will be only 60% 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 \( F = F_0e^{-0.052t} \) and we want to find \( t \) when \( F = 0.60F_0 \). So we substitute \( F \) with \( 0.60F_0 \) in the equation:
\( 0.60F_0 = F_0e^{-0.052t} \)
Since \( F_0
eq0 \) (because it's the present acreage of forest land), we can divide both sides by \( F_0 \):
\( 0.60 = e^{-0.052t} \)

Step2: Take the natural logarithm of both sides

To solve for \( t \), we take the natural logarithm (ln) of both sides. Recall that \( \ln(e^x)=x \). So:
\( \ln(0.60)=\ln(e^{-0.052t}) \)
\( \ln(0.60)= - 0.052t \)

Step3: Solve for \( t \)

Now we solve for \( t \) by dividing both sides by \( - 0.052 \):
\( t=\frac{\ln(0.60)}{- 0.052} \)
Calculate \( \ln(0.60)\approx - 0.5108 \), then:
\( t=\frac{- 0.5108}{- 0.052}\approx9.8 \)

Answer:

9.8