Question 1142904: The linear profit equation for a company that sells customized furniture is 𝑃 = 𝑎 + 𝑏𝑥, where P is the profit in dollars, x is the number of units sold, and a and b are constants. When 120 units are sold, the profit is $21000 and when 350 units are sold, the profit is $90 000.If the fixed cost is $2500 and the variable cost is $50 per unit, determine the break-even value.
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39630)   (Show Source):
Answer by ikleyn(52884)  (Show Source):
You can put this solution on YOUR website! .
The linear profit equation for a company that sells customized furniture is 𝑃 = 𝑎 + 𝑏𝑥, where P is the profit in dollars,
x is the number of units sold, and a and b are constants. When 120 units are sold, the profit is $21000
and when 350 units are sold, the profit is $90000. If the fixed cost is $2500 and the variable cost is $50 per unit,
determine the break-even value.
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This problem is, from one side, very simple.
From the other side, it is very twisted, and I will explain it to you below.
From the condition, you can determine the unknown constants " a " and "b" of the profit function in a very simple manner
(actually, MENTALLY).
The difference in units sold, 350-120 = 230, produces the difference in the profit of $90000 - $21000 = $69000.
Therefore, one produced and sold unit in this range gives the profit of  = 300 dollars.
It means that the coefficient "b" in P = a + bx is equal to 300.
Then for the coefficient "a" you have an equation
21000 = a + 300*120,
which gives you
a = 21000 - 300*120 = -15000.
Thus you restored the profit function as
P(x) = - 15000 + 300*x.
Now, the break-even value is the value of produced and sold units "x" when the profit is 0 (zero, ZERO) :
P(x) = 0, or, equivalently, -15000 + 300*x = 0,
which gives you
x =  = 50.
It is the ANSWER to the problem question:
ANSWER. The break-even value of "x" is 50.
To this point, the solution is SIMPLE and straightforward.
Now, why I say that it is twisted ?
It is because all the info related to the cost IS IRRELEVANT in this problem.
The break-even condition is P(x) = 0 in this case.
IT IS NOT P(x) = C(x), as @josgarithmetic tries to sell you.
BE AWARE : his interpretation is WRONG and can lead you to WRONG CONCLUSIONS.
This part of the condition related to the cost, is absolutely irrelevant in this problem,
and it is why I called this problem "twisted".
I don't know for what reason the part related to the cost is included to the problem.
It may happen that the author of the problem wants to check your knowledge.
It also may happen that the author of the problem does understand nothing of all what he (or she) is writing.
In the modern Internet, EVERYTHING may happen, and I am not responsible for it.
At this point, I complete my solution and my explanations.
Hope, everything is clear to you now.
Again, IGNORE THE SOLUTION by @josgarithmetic for your safety, since it is TOTALLY WRONG.
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