Question

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 linear function of the number of chairs is\(y = 13x + 900\).

(b) Theslope of the graph which represents the cost (in dollars) of producing each additional chair.

(c) \(y - {\rm{intercept}}\) is represents the fixed daily costs.

Step-by-step solution

Given data

Let\(x\)be the number of chairs produced in one day and\(y\)be the cost (in dollars)

We have been given when\(x = 100,y = 2200\)and when\(x = 300,\;y = 4800\).

So, we have two points on the graph of a linear function \(\left( {100,2200} \right)\) and\(\left( {300,4800} \right)\).

(a) Find the cost as a function of the number of chairs produced

The slope of the graph,

\(\begin{array}{c}m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\\ = \frac{{4800 - 2200}}{{300 - 100}}\\ = 13\end{array}\)

Then the equation of the linear function is;

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

Hence, the linear function is\(y = 13x + 900\).

The graph is given below:

(b) Find the slope of the graph and what does it represent

The slope of the graph is 13, which represents the cost (in dollars) of producing each additional chair.

(c) Find the y-intercept of the graph and what does it represent

The\(y - {\rm{intercept}}\)is 900 which represents the fixed daily cost.