Question

19. The manager of a furniture factory finds that it costs \(2200 to manufacture 100 chairs in one day and \)4800 to produce 300 chairs in one day.

(a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph.

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

(c) What is the y-intercept of the graph and what does it represent?

Short answer

a) The cost function is \(y = {\rm{ }}13x + {\rm{ }}900\). It is sketched as shown below:

b) The slope of the cost function is 13. It represents that increase in the cost of manufacturing an additional chair is 13 dollars.

c) The \(y\)-intercept is 900. It represents that the cost of producing zero chairs is not zero but $900.

Step-by-step solution

Determine the slope

Let the number of chairs produced in a day be \(x\) , and their cost be \(y\). It takes $2200 and $4800 to manufacture 100 and 300 chairs, respectively.

So, the points (100, 2200) and (300, 4800) must satisfy the function \(y = f\left( x \right)\). Thus the slope of the line can be calculated as:

\(\begin{aligned}\frac{{4800 - 2200}}{{{\rm{ }}300 - 100{\rm{ }}}} &= \frac{{2600}}{{200}}\\ &= 13\end{aligned}\)

Thus, the slope is 13.

Use the slope point form of a line to find the cost function

The cost function has a slope of 13 and satisfies the point (100, 2200).

According to the slope point form, the cost function is shown:

\(\begin{aligned}y - 2200 &= 13\left( {x - 100} \right)\\y &= 13x + 2200 - {\rm{13}}00\\y &= 13x + 900\end{aligned}\)

So, the cost function is given by the equation \(y = {\rm{ }}13x + {\rm{ }}900\). It is graphed as follows:

Interpret the slope of the cost function

The slope of the cost function is 13. It represents that increase in the cost of manufacturing an additional chair is 13 dollars.

Determine the y-intercept and interpret it

Observing the graph carefully shows that the line of cost function intersects the \(y\)-axis at 900. So, the \(y\)-intercept is 900.

It represents that the cost of producing zero chairs is not zero but $900.