Question

a company makes and sells skateboards. each skateboard is sold for $170. the cost to make each skateboard is $45 plus a fixed cost of $2,875 per month.

• on a monthly basis, what is the breakeven point? explain how you determined your answer.
• what does the breakeven point mean in this context?
• explain how to justify that the breakeven point is correct.

Explanation:

Step1: Set up revenue and cost functions

Let \(x\) be the number of skateboards.
Revenue function \(R(x)=170x\) (since each skateboard is sold for \(\$170\)).
Cost function \(C(x)=45x + 2875\) (variable cost \(\$45\) per skateboard and fixed cost \(\$2875\)).

Step2: Find break - even point

At break - even, \(R(x)=C(x)\).

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Answer:

• Break - even point: The break - even point is \(23\) skateboards. We set the revenue function \(R(x)=170x\) equal to the cost function \(C(x)=45x + 2875\) and solved for \(x\).
• Meaning of break - even point: It means that when the company sells \(23\) skateboards in a month, the total revenue earned from selling the skateboards is equal to the total cost of producing them. So, the company neither makes a profit nor incurs a loss.
• Justification: Substitute \(x = 23\) into both the revenue and cost functions.
• \(R(23)=170\times23=\$3910\)
• \(C(23)=45\times23+2875=1035 + 2875=\$3910\)

Since \(R(23)=C(23)\), the break - even point \(x = 23\) is correct.