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 bicycle.
If \[x\] is the number of bicycles produced and sold, then the cost function of the company is written as:
\[C\left( x \right)=100,000+100x\]
Thus, the cost function of the company is\[C\left( x \right)=100,000+100x\].
(b)
The revenue function of the company can be written as a product of price of a single bicycle and the total number of bicycles produced and sold.
If \[x\] is the number of bicycles produced and sold, then the revenue function of the company is written as\[R\left( x \right)=300x\]
The revenue function of the company is \[R\left( x \right)=300x\].
(c)
Consider the formula for cost function calculated in part (a) and revenue function calculated in part (b).
The no. of bicycles for which the cost function and revenue function are equal is known as the break-even point.
The break-even point is found by the intersection of the two functions.
\[\begin{align}
& 300x=100,000+100x \\
& 200x=100,000 \\
& x=\frac{100,000}{200} \\
& x=500
\end{align}\]
Thus, the break-even point for the company is \[500\]bicycles.
Back – substitution 500 for x in either of the system’s equations.
Now,
\[\begin{align}
& R(500)=300(500) \\
& =150000 \\
\end{align}\]
The break – even point is \[(500,150000)\].
This means that the company will break even if it produces and sells 500 bicycles. At this level,
the money coming in is equal to the money going out: \[\$150,000\].
