Question

A company manufactures small refrigerators. The total cost to manufacture 43 refrigerators is $5075. The total costs to manufacture 118 refrigerators is $10700. Assume that total cost, C, is linearly related to the number of refrigerators, x, the company manufactures and includes a fixed cost and a cost per refrigerator.

Answer 1

A) Let C(x) the cost of manufacturing x refrigerators, since C is linearly related to x, then we can set the following equation:

`C(x)=mx+b\text{.}`

To determine the values of m and b we will use the fact that:

`\begin{gathered} C(43)=5075, \\ C(118)=10700. \end{gathered}`

Therefore we can set the following system of equations:

`\begin{gathered} m\cdot43+b=5075, \\ m\cdot118+b=10700. \end{gathered}`

Subtracting the second equation from the first one we get:

`\begin{gathered} m\cdot118+b-m\cdot43-b=10700-5075, \\ 75m=5625. \end{gathered}`

Dividing the above equation by 75 we get:

`\begin{gathered} (75m)/(75)=(5625)/(75), \\ m=75. \end{gathered}`

Substituting m=75 in 43m+b=5075, and solving for b we get:

`\begin{gathered} 43\cdot75+b=5075, \\ b=5075-43\cdot75, \\ b=1850. \end{gathered}`

Therefore:

`C(x)=75x+1850.`

B) Evaluating C(x) at x=123 we get:

`\begin{gathered} C(123)=75\cdot123+1850, \\ C(123)=11075. \end{gathered}`

C) From the equation we get that the fixed cost is $1850.

D) The cost to produce each additional refrigerator is $75.

Answer:

(a)

`C(x)=75x+1850.`

(b) $11075.

(c) $1850.

(d) $75.