Chapter 3.4, Problem 83E

To determine

Find that how many years will it take to reach the population of 78

Answer to Problem 83E

The number of years it will take to reach the population of 78 will be 5 years

Explanation of Solution

Given:

The population of the beavers after x years can be approximated by

  $N=5.5\times {10}^{0.23x}0\le x\le 10$ Where $x$ is the number of years and $N$ is the population of beavers

The value of N which is $N=78$

Now, simply put the value of $N$ in the given model

  $\begin{array}{l}N=5.5\times {10}^{0.23x}\\ 78=5.5\times {10}^{0.23x}\end{array}$

  $\frac{78}{5.5}={10}^{0.23x}$ (Divide by 5.5 both the sides)

  $14.182={10}^{0.23x}$

  $\begin{array}{l}{\log }_{10}(14.182)={\log }_{10}{10}^{0.23x}(\text{ Take }\log (\text{ base }10)\text{ both the sides })\\ {\log }_{10}(14.182)=0.23x\left(\text{ since },{\log }_x{a}^2=x\right)\\ 1.1517=0.23x\\ \frac{1.1517}{0.23}=x(\text{Divide by}0.23\text{both the sides})\\ 5.007=x\end{array}$

Therefore, the number of years it will take to reach the population of 78 will be 5 years