Q 81. - Calculus

Question

In 1960, H. von Foerster suggested that the human population could be measured by the function $P\left(t\right)=\frac{179\times {10}^9}{{\left(2027-t\right)}^{0.99}}$

The time t is measured in years, where t = 1 corresponds to the year 1 a.d., 

t = 1973 corresponds to the year 1973 a.d., and so on. (We saw this “doomsday model” for population in Problem 77 of Section 1.1, on page 89.) Use limit

techniques to calculate $\underset{t\to {2027}^{-}}{\lim }P\left(t\right)$. What does this limit mean in real-world terms?

Step-by-Step Solution

Answer

Answer is $\infty$

Step 1. Given information

Human population could be measured by the function $P\left(t\right)=\frac{179\times {10}^9}{{\left(2027-t\right)}^{0.99}}$

Step 2. Explanation

Put $t=2027$ in the function width="144" style="max-width: none; vertical-align: -19px;" $P\left(t\right)=\frac{179\times {10}^9}{{\left(2027-t\right)}^{0.99}}$

$\begin{array}{rcl}\underset{t\to {2027}^{-}}{\lim }P\left(t\right)& =& \underset{t\to {2027}^{-}}{\lim }\frac{179\times {10}^9}{{\left(2027-t\right)}^{0.99}}\\ & =& \frac{179\times {10}^9}{{\left(2027-2027\right)}^{0.99}}\\ & =& \frac{179\times {10}^9}{{\left(0\right)}^{0.99}}\\ & =& \infty \end{array}$