A pair of rabbits are introduced on a small island, and the population grows until the food supply and natural enemies of the rabbits on the island limit the population. If the number of rabbits is $\mathrm { N } = \frac { 200 } { 1 + 75 \left( 2 ^ { - 0.7 t } \right) }$ where t
Is the number of years after the rabbits are introduced, how long does it take for the rabbit population to reach
100?
A) 9 years
B) 7 years
C) 10 years
D) 11 years

Question

Quizplus · Accepted Answer

The rabbit population reaches 100 when $\mathrm { N } = 100$. So, we solve the equation $\frac { 200 } { 1 + 75 \left( 2 ^ { - 0.7 t } \right) } = 100$ for t.$\frac { 200 } { 1 + 75 \left( 2 ^ { - 0.7 t } \right) } = 100$$1 + 75 \left( 2 ^ { - 0.7 t } \right) = 2$$75 \left( 2 ^ { - 0.7 t } \right) = 1$$2 ^ { - 0.7 t } = \frac { 1 } { 75 }$$- 0.7 t = \log _ { 2 } \left( \frac { 1 } { 75 } \right)$$t = \frac { \log _ { 2 } \left( \frac { 1 } { 75 } \right) } { - 0.7 } \approx 9$So, it takes about 9 years for the rabbit population to reach 100.

A Pair of Rabbits Are Introduced on a Small Island N=2001+75(2−0.7t)\mathrm { N } = \frac { 200 } { 1 + 75 \left( 2 ^ { - 0.7 t } \right) }N=1+75(2−0.7t)200​

A Pair of Rabbits Are Introduced on a Small Island $\mathrm { N } = \frac { 200 } { 1 + 75 \left( 2 ^ { - 0.7 t } \right) }$