Chapter 0.3, Problem 20E

a

To determine

The population in year 1915 and 1940.

Answer to Problem 20E

The population in 1915 and 1940 are 12315 and 24265 respectively.

Explanation of Solution

Given:

The population in 1890 is 6250.

And the increasing rate is 2.75% per year

Concept Used:

The general formula for the exponential model is:

  $P\left(t\right)=P_0{\left(1+r\right)}^t$

Where,

  $P_0\to$ Initial amount

  $P\left(t\right)\to$ Final amount

  $r\to$ Rate

  $t\to$ Time

Calculation:

Here, the year 1890 denotes $t=0$ i.e. $P_0=6250$ .

The value of $t$ for year 1915 is:

  $\begin{array}{l}t=1915-1890\\ t=25\end{array}$

The population for 25 years is:

  $\begin{array}{c}P\left(t\right)=6250{\left(1+\frac{2.75}{100}\right)}^{25}\\ P\left(t\right)=6250{\left(1.0275\right)}^{25}\\ P\left(t\right)\approx 12315\end{array}$

The value of $t$ for year 1940 is:

  $\begin{array}{l}t=1940-1890\\ t=50\end{array}$

The population for 50 years is:

  $\begin{array}{c}P\left(t\right)=6250{\left(1+\frac{2.75}{100}\right)}^{50}\\ P\left(t\right)=6250{\left(1.0275\right)}^{50}\\ P\left(t\right)\approx 24265\end{array}$

Hence, the population in 1915 and 1940 are 12315 and 24265 respectively.

b

To determine

When the population reach 50,000.

Answer to Problem 20E

At 76.7 years the population will reach 50000.

Explanation of Solution

Given:

  $\begin{array}{l}P\left(t\right)=50000\\ P_0=6250\\ r=2.75\%\end{array}$

Concept Used:

The general formula for the exponential model is:

  $P\left(t\right)=P_0{\left(1+r\right)}^t$

Where,

  $P_0\to$ Initial amount

  $P\left(t\right)\to$ Final amount

  $r\to$ Rate

  $t\to$ Time

Calculation:

Substitute the values in the formula:

  $\begin{array}{l}50000=6250{\left(1+\frac{2.75}{100}\right)}^t\\ \frac{50000}{6250}={\left(1.0275\right)}^t\end{array}$

Take natural logarithm to both sides:

  $\begin{array}{l}50000=6250{\left(1+\frac{2.75}{100}\right)}^t\\ \ln \left(\frac{50000}{6250}\right)=t\ln \left(1.0275\right)\\ t=\frac{\ln \left(\frac{50000}{6250}\right)}{\ln \left(1.0275\right)}\\ t\approx 76.7\end{array}$

Hence, At 76.7 years the population will reach 50000.