Provide an appropriate response.
-Urban encroachment is causing the area of a forest to decline at a rate of 6% per year. Use the approximate half-life formula to determine the fraction that remains in 66 years.
A)0.942318
B)2.828571
C)0.019817
D)0.520202

Question

Quizplus · Accepted Answer

The approximate half-life formula for a quantity decreasing at a rate of $r$ per year is given by $T_{1/2} = \frac{0.693}{r}$. For a 6% decline rate, $r = 0.06$, so $T_{1/2} = \frac{0.693}{0.06} \approx 11.55$ years. To find the fraction remaining after 66 years, we use the formula for exponential decay: $N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$, where $t = 66$ years. Substituting the values, we get $N = 1 \left(\frac{1}{2}\right)^{\frac{66}{11.55}} \approx 0.019817$, indicating that about 1.9817% of the forest remains after 66 years.

Provide an Appropriate Response