Video: Break-Even Point of a Linear Function

A guitar factory has a cost of production given by 𝐶(𝑥) = 75𝑥 + 50000. If the company needs to break even after 150 units sold, at what price should they sell each guitar? Give your answer to the nearest dollar.

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Video Transcript

A guitar factory has a cost of production given by 𝐶 of 𝑥 is equal to 75𝑥 plus 50000. If the company needs to break even after 150 units sold, at what price should they sell each guitar? Give your answer to the nearest dollar.

Well, the first thing to do is to work out the total cost of production for the 150 units sold. So therefore, using the formula we’ve been given, we can say that the cost for 150 units is gonna be equal to 75 multiplied by 150. And that’s because 𝑥 is equal to 150 because it’s 150 units and then plus 50000, which is gonna be equal to 11250 plus 50000, which would give us a total cost of 61250 dollars. Okay, great! So we’ve got the total cost to produce our 150 units.

Now, we’re told that the company needs to break even. So what this means is that no profit and no loss will be made when 150 of the units are sold. So therefore, the company wants the total cost recovered through the sales of the guitars.

So therefore, we can say that the cost per guitar is gonna be equal to 61250 divided by 150 because we’ve already said there’re 150 units to be sold, which is gonna give us an answer of 408.3 recurring. Well, we want the price to the nearest dollar. So therefore, the cost of each guitar is gonna be equal to 409 dollars.

Now I think, well, why is it 409 dollars? Surely, if it’s to the nearest dollar, it would be 408 dollars because 408.3 recurring is closer to 408. Well, the reason is because if we had 408 dollars, then 408 dollars multiplied by 150 would give us 61200 dollars. And this would not cover the costs. So not only with the not break even, they would actually be losing money.

So therefore, we’d rounded it up to 409 dollars because, therefore, we’d almost break even with a slight profit, which would be favorable for the company.

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