Single Variable Calculus, Chapter 1, Review Exercises, Section Review Exercises, Problem 22

According to the manufacturer of oven toasters, it cost $\$9,000$ to produce $1,000$ toaster ovens a week and $\$12,000$ to produce $1,500$ toasters a week.

a) Assuming that it is linear, express the cost as a function of the number of toaster ovens produced. Sketch its graph.


$
\begin{equation}
\begin{aligned}

\text{Let } C = aT + k \text{ where: } C =& \text{cost}\\
T =& \text{number of toaster produced}\\
k =& \text{constant}\\
a =& \text{slope}


\end{aligned}
\end{equation}
$


We need to find the values of $a$ and $k$ to express the function correctly and to fit with the given data.



$
\begin{equation}
\begin{aligned}

\text{when } C =& \$9,000 \quad \text{ and } \quad T = 1,000\\
900 = & a(1000) + k\\
1000a + k =& 9,000
\qquad \text{Equation 1}\\
\text{when } C =& \$ 12,000 \quad \text{ and } \quad T= 1,500\\
1500a + k =& 12,000
\qquad \text{Equation 2}\\
\\
\qquad \text{ Combining Equations 1 and 2, we get }
a =& 6\\
k =& 3,000

\end{aligned}
\end{equation}
$


Therefore, the expression is $C = 6T + 3,000$







b) What is the slope of the graph and what does it represent?

The slope is 6, it represents the change of the amount as the number of toasters produced changes.



c) What is the y-intercept of the graph and what does it represent?
The $y$ intercept is 3,000, it represents the amount of charges outside production like maintenance of the equipments used or some rental charges.