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Question 1167949: The demand equation for a certain product is given by p = 118 −0.045x, where p is the unit price (in dollars) of the product and x is the number of units produced.
The total revenue obtained by producing and selling x units is given by R = x*p.
Determine prices (p) that would yield a revenue of 9200 dollars.
Lowest such price = ? dollars
Highest such price = ? dollars
Please help me with this!!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! not sure if this is right, but this is what i get.
R = x * p
p = 118 - .045x
p in both equation equals the price per unit.
if you replace p in the first equation with its equivalent value of 118 - .045x, then you get:
R = x * (118 - .045x)
simplify to get:
R = 118 * x - .045 * x^2
if you replace R with y, the equation becomes y = 118 * x - .045 * x^2
y represents the revenue, just like R represented the revenue.
this allows the equation to be graphed more easily, since a lot of graphing software assumes the vertical axis is the y-axis and the horizontal axis is the x-axis.
if we reorder the terms in descending order of degree and switch sides, the equation remains the same, but is shown as:
-.045 * x^2 + 118 * x = y
if we want the revenue to be equal to 9200, we set y equal to 9200 and we get:
-.045 * x^2 + 118 * x = 9200
if we want to solve this quadratic equation, we subtract 9200 from both sides to get:
-.045 * x^2 + 118 * x - 9200 = 0
when we solve this quadratic equation, we get:
x = 80.43328924441 or x = 2541.7889329778
those values of x make the equation true.
if we add 9200 to both sides of the equation, the equation is still true.
we get:
-.045 * x^2 + 118 * x = 9200
what all this is saying is that when x = 80.43328924441 or x = 2541.7889329778, y = 9200, which is saying that the revenue is equal to 9200.
we have values of x, but we don't have values of p.
we know that x * p = revenue, so we solve for p by dividing 9200 by x.
we get:
9200 / 80.43328924441 = 114.38050198
9200 / 2541.7889329778 = 3.61949802
i didn't round because, if you wanted to check if the arithmetic is correct, you would have to use the unrounded numbers or you would come up with some small discrepancies.
i will round now, since i showed you what the unrounded numbers are.
the number of units probably should be rounded to the nearest unit.
the money probably should be rounded to the nearest penny.
after doing that, you get:
if you produce about 80 units, the price per unit should be about 114.38.
if you produce about 2542 units, the price per unit should be about 3.62
both the demand equation and the revenue equation are satisfied when x = 80.43328924441 or x = 2541.7889329778
demand equation is p = 118 - .045 * x
when x = 80.43328924441, p = 114.38050198
when x = 2541.78893298, p = 3.61949802
revenue equation is r = x * p
when r = 9200, the equation becomes 9200 = x * p
when x = 80.43328924441, p = 114.38050198
when x = 2541.78893298, p = 3.61949802
both the demand equation and the revenue equation are satisfied with those number.
that's the best i can do.
i think it's right, but i always allow for the possibility that i could be wrong.
good luck with it.
hopefully this helps.
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