Q. 77 - Calculus

Question

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

Here P is the size of the human population. The time t is measured in years, where t = 1 corresponds to the year 1 A.D., time t = 1973 corresponds to the year 1973 A.D., and so on.

(a) Use a graphing utility to graph this function. You will have to be very careful when choosing a graphing window!

(b) Use the graph you found in part (a) to approximate $\underset{t\to {2027}^{-}}{\lim }P\left(t\right).$

(c) This population model is sometimes called the doomsday model. Why do you think this is? What year is doomsday, and why?

(d) In part (b), we considered only the left limit of P(t) as $x\to 2027$ Why? What is the real-world meaning of the part of the graph that is to the right of t = 2027?

Step-by-Step Solution

Answer

Part (a) The graph of the function by using the graphing utility is

Part (b) The approximate value of the limit $\underset{t\to {2027}^{-}}{\lim }P\left(t\right)=\infty .$

Part (c) The Doomsday is a probabilistic statement that claims to indicate the number of future members of the human species given an appraisal of the total number of humans born so far. That's why the population model is called a doomsday model.

Part (d) In part (b), we considered only the left limit of P(t) as $x\to 2027$ because the model is designed for until the year 2027. According to this model, no world would be left after 2027.

Part (a) Step 1. Given Information.

The given function is $P\left(t\right)=\frac{179\times {10}^9}{{\left(2027-t\right)}^{0.99}}.$

Part (a) Step 2. Sketch the graph of the function by using a graphing utility.

To graph Pt) by using a graphing utility, insert the function and adjust the window.

The graph is

Part (b) Step 1. Approximating lim t → 2027 - P ( t ) .

From the graph, we can depict that the value of the function at $t\to {2027}^{-}$ is $\infty .$

Therefore, the value of the limit $\underset{t\to {2027}^{-}}{\lim }P\left(t\right)=\infty .$

Part (c) Step 1. Explanation.

The Doomsday is a probabilistic statement that claims to indicate the number of future members of the human species given an appraisal of the total number of humans born so far. That's why the population model is called a doomsday model.

Part (d) Step 1. Explanation.

In part (b), we considered only the left limit of P(t) as $x\to 2027$ because the model is designed for until the year 2027. According to this model, no world would be left after 2027.