Chapter 8.8, Problem 18HP

To determine

To find: The difference between exponential, continuous exponential, logistic growth and calculate population.

Answer to Problem 18HP

The population after 30 years will be $10991.51$ .

Explanation of Solution

Given information:

Calculation:

The exponential function can be used to model situations that incorporate a percentage of growth or decay for a specific number of times per year.

Now, taking an example of an exponential function , $U-238$ is modeled by the equation is given as below,

  $y=a{e}^{-1.54\times {10}^{-10}t}$

Where, $y=$ amount of uranium after decay, $a$ is initial amount and $t$ is time in years.

There is $100$ mg of uranium initially. Then after $4.5$ millions year amount left is given as,

  $\begin{array}{c}y=a{e}^{-1.54\times {10}^{-10}t}\\ =100e^{-1.54\times {10}^{-10}\times 4.5\times {10}^9}\\ =50.007\end{array}$

Therefore, the amount of uranium left after $4.5$ billion years is $50.007$ mg.

The continuous exponential function can be used to model situations that incorporates a percentage of growth or decay continuously.

The logistic function can be used to model situations that incorporate a percentage of growth or decay continuously and consists of a limiting factor.

Consider the following logistic function.

  $f(t)=\frac{c}{1+a{e}^{-bt}}$

Consider the example of population growth. The population can be modeled using the equation

  $f(t)=\frac{18000}{1+e^{-0.015\times 30}}$

Hence, population after 30 years is

  $\begin{array}{c}f(t)=\frac{18000}{1+e^{-0.015\times 30}}\\ =\frac{18000}{1+0.638}\\ =10991.51\end{array}$

Therefore the population after 30 years will be $10991.51$