Chapter 1.2, Problem 82E

(a)

To determine

The intercepts of the red line graphically, and interpret the meaning of each intercept.

The $y$ -intercept and the $x$ -intercept of the red line is $\left(0,\text{\$}10000\right)$ and $\left(\text{\$}20000,0\right)$, respectively.

Explanation:

Consider the linear model.

For subsidy,

$S=10000-\frac{1}{2}E,\text{ 0}\le E\le 20000$

Here, $E$ represents the earned income.

For total income,

$T=E+S$

The provided graph is:

By observing the graph, it can be estimated that the red line cuts the $y$ -axis and the $x$ -axis at 10 and $20$, respectively. So, the $y$ -intercept is $\left(0,\text{\$}10000\right)$ and the $x$ -intercept is $\left(\text{\$}20000,0\right)$.

The $x$ -intercept means that the subsidy is $\text{\$}0$ for an earned income of $\text{\$}20,000$ and the $y$ -intercept, that is, $\left(0,\text{\$}10000\right)$ means that the subsidy is $\text{\$}10,000$ for $\text{\$}0$ earned income.

Hence, the $y$ -intercept and the $x$ -intercept of the red line is $\left(0,\text{\$}10000\right)$ and $\left(\text{\$}20000,0\right)$, respectively.

(b)

To determine

To calculate: The intercepts of the red line algebraically.

The $y$ -intercept and the $x$ -intercept of the red line is $\left(0,\text{\$}10000\right)$ and $\left(\text{\$}20000,0\right)$, respectively.

Calculation:

Consider the provided graph.

The red line represents,

$S=10000-\frac{1}{2}E,\text{ 0}\le E\le 20000$

Upon substituting $E=0$ and simplifying the equation,

$\begin{array}{c}S=10000-\frac{1}{2}\left(0\right)\\ =10000\end{array}$

Thus, the $y$ -intercept is $\left(0,10000\right)$.

Now, substitute $S=0$ and simplify the equation.

$\begin{array}{c}0=10000-\frac{1}{2}E\\ 0-10000=10000-\frac{1}{2}E-10000\\ -2\cdot \left(-10000\right)=-2\cdot \left(-\frac{1}{2}E\right)\\ 20000=E\end{array}$

Thus, the $x$ -intercept is $\left(20000,0\right)$.

Hence, the $y$ -intercept and the $x$ -intercept of the red line is $\left(0,\text{\$}10000\right)$ and $\left(\text{\$}20000,0\right)$, respectively.

(c)

To determine

The earned income for which the total income is $\text{\$}14000$, graphically.

The earned income for which the total income is $\text{\$}14000$ is $\text{\$}8000$.

Explanation:

Consider the provided graph.

The blue line represents the Total Income,

$T=E+S$

Observe the graph below.

Hence, the earned income for which the total income is $\text{\$}14000$ is $\text{\$}8000$.

(d)

To determine

To calculate: The earned income when the total income is $\text{\$}14000$.

The earned income for which the total income is $\text{\$}14000$ is $\text{\$}8000$.

Calculation:

Consider the linear model for total income,

$T=E+S$

Substitute $T=14000$ in the equation.

$\begin{array}{c}14000=E+10000-\frac{1}{2}E\\ 14000-10000=\frac{1}{2}E+10000-10000\\ 2\cdot 4000=2\cdot \frac{1}{2}E\\ 8000=E\end{array}$

Check:

Substitute $E=8000$ in $14000=E+10000-\frac{1}{2}E$.

$\begin{array}{l}14000\stackrel{?}{=}8000+10000-\frac{1}{2}\left(8000\right)\\ 14000\stackrel{?}{=}8000+10000-4000\\ 14000=14000\end{array}$

Thus, it is true.

Hence, the earned income for which the total income is $\text{\$}14000$ is $\text{\$}8000$.