Q 82. - Calculus

Question

Suppose instead we consider the population model

$Q\left(t\right)=\frac{44\times {10}^{10}}{1+{\left(2027-t\right)}^{{\displaystyle \frac{4}{3}}}}$

with t measured in years as in the previous problem.

(a) Use limit techniques to calculate $\underset{t\to \infty }{\lim }Q\left(t\right)$. What does this limit mean in real–world terms? What happens in this model in the year 2027?

(b) Use calculator graphs to compare the population models in this exercise with those in the previous exercise. Describe the long–term population growth

scenarios that are suggested by these models.

Step-by-Step Solution

Answer

(a) Answer is 0.

(b) Answer is $44\times {10}^{10}$

Step 1. Given information

The population model is $Q\left(t\right)=\frac{44\times {10}^{10}}{1+{\left(2027-t\right)}^{{\displaystyle \frac{4}{3}}}}$

Part (a) Step 1. Explanation

$\begin{array}{rcl}\underset{t\to \infty }{\lim }Q\left(t\right)& =& \underset{t\to \infty }{\lim }\left[\frac{44\times {10}^{10}}{1+{\left(2027-t\right)}^{{\displaystyle \frac{4}{3}}}}\right]\\ & =& \frac{44\times {10}^{10}}{1+{\left(2027-\infty \right)}^{{\displaystyle \frac{4}{3}}}}\\ & =& \frac{44\times {10}^{10}}{\infty }\\ & =& 0\end{array}$

Part (b) Step 1. Explanation

$\begin{array}{rcl}\underset{t\to {2027}^{-}}{\lim }Q\left(t\right)& =& \underset{t\to {2027}^{-}}{\lim }\left[\frac{44\times {10}^{10}}{1+{\left(2027-t\right)}^{{\displaystyle \frac{4}{3}}}}\right]\\ & =& \frac{44\times {10}^{10}}{1+{\left(2027-2027\right)}^{{\displaystyle \frac{4}{3}}}}\\ & =& \frac{44\times {10}^{10}}{1+0}\\ & =& 44\times {10}^{10}\end{array}$