The weekly cost of manufacturing x telephones per week is found by a manufacturer to be C(x)= 500+20x+xsq dollars.
The telephones can be sold at a price p=$80 each.
a)find the manufacturers's break-even production level(s)
b)for what production levels will the manufacturer experience a profit?
The is 218
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The answer is 218 people
I'm still unsure what question or problem you are referring to, but if the answer is 218 people, then you may have been asked to determine a certain quantity or number of people in a given scenario or context.
Without more information, it's difficult to provide further insight or explanation. Please provide more context or details about the question or problem you are referring to so I can better assist you.
To find the manufacturer's break-even production level(s), we need to determine the production level(s) at which the cost equals the revenue (no profit or loss).
a) To find the break-even production level(s), we set the cost equal to the revenue:
C(x) = p * x
Where:
C(x) = Cost function
p = Sale price of each telephone
x = Production level
In this case, the cost function is given by C(x) = 500 + 20x + xsq dollars and the sale price of each telephone is p = $80.
500 + 20x + xsq = 80x
To simplify the equation, we rearrange it:
xsq - 60x + 500 = 0
This is a quadratic equation, which can be solved using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in the values, we have:
x = (-(-60) ± sqrt((-60)^2 - 4(1)(500))) / (2(1))
x = (60 ± sqrt(3600 - 2000)) / 2
x = (60 ± sqrt(1600)) / 2
x = (60 ± 40) / 2
This gives us two possible break-even production levels:
x1 = (60 + 40) / 2 = 50
x2 = (60 - 40) / 2 = 10
Therefore, the manufacturer's break-even production levels are 50 and 10 telephones per week.
b) To find the production levels at which the manufacturer will experience a profit, we need to consider the revenue being greater than the cost.
Since the sale price of each telephone is $80, the revenue function is given by:
R(x) = p * x = 80x
To find the production levels that yield a profit, we need to compare the revenue with the cost:
R(x) > C(x)
80x > 500 + 20x + xsq
Rearranging the equation:
xsq + 60x - 500 > 0
Again, this is a quadratic equation. We need to find the values of x that satisfy this inequality. To determine these production levels, we find the x-intercepts of the quadratic equation:
xsq + 60x - 500 = 0
Using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in the values, we have:
x = (-60 ± sqrt((60)^2 - 4(1)(-500))) / (2(1))
x = (-60 ± sqrt(3600 + 2000)) / 2
x = (-60 ± sqrt(5600)) / 2
x = (-60 ± 74.83) / 2
This gives us two possible ranges of production levels that will result in a profit:
-67.42 < x < 7.42
Therefore, the manufacturer will experience a profit for production levels between approximately -67.42 and 7.42 telephones per week.