Answer to Question #105038 in Algebra for Bayrhuu

Question

10.Mebel Factory Manager knows that it costs $ 2200 per 100 chairs and $ 4800 per 300 chairs.
(a) Express the cost as a function of the number of chairs to produce. In doing so, consider the function linear.
(b) Find the slope of the graph. What does this mean?
(c) Find the value of intersecting the y-axis and explain what it means.

Answer 1

Question (a)

Let the cost function be

$$y(x)=kx+b,\,\,\,\text{where}\,\,\,x\,\,\,\text{ is the number of chairs}$$

To determine the constants, we use the conditions of the problem

$$\left\{\begin{array}{l} y(100)=2200=100k+b\\ y(300)=4800=300k+b \end{array}\right.\rightarrow\\[0.3cm] \left\{\begin{array}{l} 2200=100k+b\\ 4800-2200=(300k+b)-(100k+b) \end{array}\right.\rightarrow\\[0.3cm] \left\{\begin{array}{l} 2200=100k+b\\ 2600=200k\to k=\displaystyle\frac{2600}{200}=13 \end{array}\right.\rightarrow\\[0.3cm] \left\{\begin{array}{l} 2200=100\cdot 13+b\to b=900\\ k=13 \end{array}\right.\rightarrow \boxed{\left\{\begin{array}{l} b=900\\ k=13 \end{array}\right.}\\[0.3cm]$$

Conclusion,

$$\boxed{y(x)=13x+900-\text{cost function}}$$

Question(b)

The slope of this function is

$$\boxed{k=13}$$

This ratio means that for every chair sold we get $13.

Question(c)

The intersection with the $Oy-$ axis is

$$y(0)=13\cdot 0 +900\longrightarrow\boxed{y(0)=900}$$

This number shows fixed production costs such as: rental of premises; salary; rent / purchase of equipment and materials, etc.

ANSWER

(a)

$$y(x)=13x+900-\text{cost function}$$

(b)

The slope of this function is

$$k=13$$

This ratio means that for every chair sold we get $ 13.

(c)

The intersection with the Oy− axis is

$$y(0)=900$$

This number shows fixed production costs such as: rental of premises; salary; rent / purchase of equipment and materials, etc.