Chapter 1.2, Problem 64E

To determine

To find the formula for the total mass, sketch graphs of $P\left(t\right),$ the mass per person, the total mass as functions of time for $0\le t\le 100$ , and describe the results.

Answer to Problem 64E

Hence the graph is drawn and the total mass is $T\left(t\right)=\left(2.0\times {10}^6+2.0\times {10}^{4}t\right).\left(80+0.5t\right)$

Explanation of Solution

Given information:

The population of people is

  $P\left(t\right)=2.0\times {10}^6+2.0\times {10}^{4}t\mathrm{.............}\left(A\right)$

The mass per person $W\left(t\right)$ (in kg) is

  $W\left(t\right)=80+0.5t\mathrm{.......}\left(B\right)$

Again, the total mass $T\left(t\right)$ of the population is

  $T\left(t\right)=P\left(t\right).W\left(t\right)\mathrm{........}\left(C\right)$

Hence, the total mass $T\left(t\right)$ of the population is

  $T\left(t\right)=\left(2.0\times {10}^6+2.0\times {10}^{4}t\right).\left(80+0.5t\right)$

Calculation:

Now, we need to find the values of $P\left(t\right)$ at $t=\mathrm{0.0.20.0.40.0.60.0.80.0.100.0}$

At $t=0.0$ , we have

  $\begin{array}{l}P\left(0.0\right)=2.0\times {10}^6+2.0\times {10}^4\left(0.0\right)\\ =2.0\times {10}^6+0.0\\ =2.0\times {10}^6\\ P\left(0.0\right)=2.0\times {10}^6\end{array}$

At $t=20.0$ , we have

  $\begin{array}{l}P\left(20.0\right)=2.0\times {10}^6+2.0\times {10}^4\left(20.0\right)\\ =2.0\times {10}^6+0.4\times {10}^6\\ =2.4\times {10}^6\\ P\left(20.0\right)=2.4\times {10}^6\end{array}$

At $t=40.0$ , we have

  $\begin{array}{l}P\left(40.0\right)=2.0\times {10}^6+2.0\times {10}^4\left(40.0\right)\\ =2.0\times {10}^6+0.8\times {10}^6\\ =2.8\times {10}^6\\ P\left(40.0\right)=2.8\times {10}^6\end{array}$

At $t=60.0$ , we have

  $\begin{array}{l}P\left(60.0\right)=2.0\times {10}^6+2.0\times {10}^4\left(60.0\right)\\ =2.0\times {10}^6+1.2\times {10}^6\\ =3.2\times {10}^6\\ P\left(60.0\right)=3.2\times {10}^6\end{array}$

At $t=80.0$ , we have

  $\begin{array}{l}P\left(80.0\right)=2.0\times {10}^6+2.0\times {10}^4\left(80.0\right)\\ =2.0\times {10}^6+1.6\times {10}^6\\ =3.6\times {10}^6\\ P\left(80.0\right)=3.6\times {10}^6\end{array}$

At $t=100.0$ , we have

  $\begin{array}{l}P\left(100.0\right)=2.0\times {10}^6+2.0\times {10}^4\left(100.0\right)\\ =2.0\times {10}^6+2.0\times {10}^6\\ =4.0\times {10}^6\\ P\left(100.0\right)=4.0\times {10}^6\end{array}$

Now, we need to find the values of $W\left(t\right)$ at $t=\mathrm{0.0.20.0.40.0.60.0.80.0.100.0}$

At $t=0.0$ , we have

  $\begin{array}{l}W\left(0.0\right)=80+0.5\cdot 0.0\\ =80+0.0\\ =80\\ W\left(0.0\right)=80\end{array}$

At $t=20.0$ , we have

  $\begin{array}{l}W\left(20.0\right)=80+0.5\cdot 20.0\\ =80+10.0\\ =90\\ W\left(20.0\right)=90\end{array}$

At $t=40.0$ , we have

  $\begin{array}{l}W\left(40.0\right)=80+0.5\cdot 40.0\\ =80+20.0\\ =100\\ W\left(40.0\right)=100\end{array}$

At $t=60.0$ , we have

  $\begin{array}{l}W\left(60.0\right)=80+0.5\cdot 60.0\\ =80+30.0\\ =110\\ W\left(60.0\right)=110\end{array}$

At $t=80.0$ , we have

  $\begin{array}{l}W\left(80.0\right)=80+0.5\cdot 80.0\\ =80+40.0\\ =120\\ W\left(80.0\right)=120\end{array}$

At $t=100.0$ , we have

  $\begin{array}{l}W\left(100.0\right)=80+0.5\cdot 100.0\\ =80+50.0\\ =130\\ W\left(100.0\right)=130\end{array}$

Now, we need to find the values of $T\left(t\right)$ at $t=0.0,0.5,1.0,1.5,2.0,2.5,3.0$

At $t=0.0$ , we have

  $\begin{array}{l}T\left(0.0\right)=P\left(0.0\right).W\left(0.0\right)\\ =\left(2.0\times {10}^6\right)\cdot 80.0\\ =160.0\times {10}^6\\ T\left(0.0\right)=160.0\times {10}^6\end{array}$

At $t=20.0$ , we have

  $\begin{array}{l}T\left(20.0\right)=P\left(20.0\right).W\left(20.0\right)\\ =\left(2.4\times {10}^6\right)\cdot 90.0\\ =216.0\times {10}^6\\ T\left(20.0\right)=216.0\times {10}^6\end{array}$

At $t=40.0$ , we have

  $\begin{array}{l}T\left(40.0\right)=P\left(40.0\right).W\left(40.0\right)\\ =\left(2.8\times {10}^6\right)\cdot 100.0\\ =280.0\times {10}^6\\ T\left(40.0\right)=280.0\times {10}^6\end{array}$

At $t=60.0$ , we have

  $\begin{array}{l}T\left(60.0\right)=P\left(60.0\right).W\left(60.0\right)\\ =\left(3.2\times {10}^6\right)\cdot 110.0\\ =352.0\times {10}^6\\ T\left(60.0\right)=352.0\times {10}^6\end{array}$

At $t=80.0$ , we have

  $\begin{array}{l}T\left(80.0\right)=P\left(80.0\right).W\left(80.0\right)\\ =\left(3.6\times {10}^6\right)\cdot 120.0\\ =432.0\times {10}^6\\ T\left(80.0\right)=432.0\times {10}^6\end{array}$

At $t=100.0$ , we have

  $\begin{array}{l}T\left(100.0\right)=P\left(100.0\right).W\left(100.0\right)\\ =\left(4.0\times {10}^6\right)\cdot 130.0\\ =520.0\times {10}^6\\ T\left(100.0\right)=520.0\times {10}^6\end{array}$

The graph for $P\left(t\right)$ is shown below.

  

The graph for $W\left(t\right)$ is shown below.

  

The graph for $T\left(t\right)$ is shown below.

  

We can see from the graphs above that the population increases, the mass per individual increases and the total mass also increases.