Chapter 9.6, Problem 58E

(a)

To determine

To Find: The time when the country will experience the food shortage.

Answer to Problem 58E

The shortage will occur after 84 days.

Explanation of Solution

Given:

The population of the country is 2 million people and the population increase by 3% each year.

The country food supply is sufficient to provide 3 million people and it increase at the constant rate that feed 0.25 million additional people each year.

Calculation:

Consider the exponential growth model for the population is,

$y=a{\left(1+r\right)}^t$

Then,

$\begin{array}{c}y=2{\left(1+0.03\right)}^t\\ =2{\left(1.03\right)}^t\end{array}$

Since, the food supply is the linear function where $m$ is 0.25. Then, use the slope intercept form as,

$y=0.25t+3$

Then solve as,

$\begin{array}{l}2{\left(1.03\right)}^t=0.25t+3\\ 2{\left(1.03\right)}^t-0.25t-3=0\end{array}$

Then the function is $f\left(x\right)=2{\left(1.03\right)}^t-0.25t-3$ .

Then, by the help of the graphing utility draw the graph as shown in Figure 1

  

Figure 1

The graph shown that the shortage will occur after 84 days.

(b)

To determine

To Find: Whether the food shortage will occur when the country doubles the rate at which the food is supplied and then find the year if it is.

Answer to Problem 58E

The shortage will occur after 115 days.

Explanation of Solution

Calculation:

Consider the required linear model is,

$y=0.5t+3$

Then, the rate is doubled as shown in Figure 2

  

Figure 2

From the graph the shortage will occur after 115 days.