Answer
See below:
Work Step by Step
(a)
The cost function of the company can be written as a sum of the fixed cost of the company and the cost to produce each canoe.
If \[x\] is the number of canoes produced and sold, then the cost function of the company is written as:
\[C(x)=18,000+20x\]
Thus, the cost function of the company is\[C\left( x \right)=18,000+20x\].
(b)
The revenue function of the company can be written as a product of price of a single canoe and the total number of canoes produced and sold.
If \[x\] is the number of canoes produced and sold, then the revenue function of the company is written as\[R\left( x \right)=80x\].
Thus, the revenue function of the company is\[R\left( x \right)=80x\].
(c)
Consider the formula for cost function calculated in part (a) and revenue function calculated in part (b).
The no. of canoes for which the cost function and revenue function are equal is known as the break-even point.
The break-even point is found by equating both the functions equal such as:
\[\begin{align}
& 80x=18,000+20x \\
& 60x=18,000 \\
& x=\frac{18,000}{60} \\
& x=300
\end{align}\]
Back – substitution 300 for x in either of the system’s equations:
\[C(x)=18,000+20x\text{ and }R(x)=80x\]
Now,
\[\begin{align}
& R(300)=80(300) \\
& =24000 \\
\end{align}\]
The break – even point is \[(300,24000)\].
This means that the company will break even if it produces and sells 300 canes. At this level,
the money coming in is equal to the money going out: \[\$24000\].
