Question 1104108
<font color="black" face="times" size="3">
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 <font color=red>$57.14</font>


----------------------------------------------------------------------


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)

</font>