SOLUTION: The demand curve for a product is given by q=800-7p^2 , where p is the price. Find the price that maximizes revenue for sales of this product.

Question 1104108: The demand curve for a product is given by q=800-7p^2 , where p is the price. Find the price that maximizes revenue for sales of this product.

Found 3 solutions by rothauserc, jim_thompson5910, josgarithmetic:
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
The demand curve is a parabola that curves downward, therefore the price (x axis) that maximizes the revenue is the x coordinate value for the parabola's vertex
:
q = 800 - 7p^2
:
x coordinate of the vertex is -b/2a = 0 / 2(-7) = 0
:
here is a graph of the demand curve
:

:

Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

demand equation: q=800-7p^2
q = number of items demanded (aka number of items willing to be bought)
p = price per item in dollars

Revenue = (price per item)*(number of items bought)
R = (p)*(q)
R = p*(800-7p^2)
R = p*800+p*(-7p^2)
R = 800p-7p^2
R = -7p^2+800p

Let f(x) = -7x^2+800x. Finding the vertex of f(x) will lead to the max value of R(p)

In the case of f(x) = -7x^2+800x, it is in the form f(x) = ax^2+bx+c. So a = -7, b = 800 and c = 0

Vertex = (h,k)

Use this formula
h = -b/(2*a)
to find the x coordinate of the vertex

h = -b/(2*a)
h = -800/(2*(-7)) <<---- plugging in a = -7 and b = 800
h = -800/(-14)
h = 800/14
h = 400/7
h = 57.1428571428571 <<---- this value is approximate
h = 57.14 <<---- rounding to 2 decimal places (aka to the nearest penny)

The x coordinate of the vertex is roughly 57.14. Since I replaced p with x, this indicates that the x coordinate of the vertex corresponds to the p coordinate of (p, R(p))

So the max revenue will occur when the price per item is $57.14

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Extra Info:

Plug the value of h into the f(x) function to get the y coordinate of the vertex
This will give the max revenue
k = f(h)
k = -7(57.14)^2+800(57.14)
k = 22,857.1428
k = 22,857.14
The y coordinate of the vertex is 22,857.14 indicating the largest revenue possible is $22,857.14 (which happens when the price per item is set at $57.14)

Answer by josgarithmetic(39623)   (Show Source): You can put this solution on YOUR website!

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