SOLUTION: A shoe store sells a new model of athletic shoe for $90. On average, the store sells 45 pairs of these shoes each week. Jason, the sales manager, estimates that he will sell 4 addi

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Question 960240: A shoe store sells a new model of athletic shoe for $90. On average, the store sells 45 pairs of these shoes each week. Jason, the sales manager, estimates that he will sell 4 additional pairs of shoes each week for every $5 reduction in the price.
Find the following:
1.) A quadratic function modeling the store's average weekly revenue from the sales of these shoes as related to reduction in price.
2.) The range of prices which would allow the store to maintain or increase its current weekly revenue from these shoes.
3.) The maximum amount by which the store can increase its revenue from these shoes each week. Round off to the nearest whole dollar.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +R+ = average weekly revenue
Let +n+ = number of $5 reductions in the price of shoes
--------------------------
(1)
+R+=+%28+45+%2B+4n+%29%2A%28+90+-+5n+%29+
+R+=+4050+%2B+360n+-+225n+-+20n%5E2+
+R+=+-20n%5E2+%2B+135n+%2B+4050+
-------------------------------
(2)
Current weekly revenue is +45%2A90+=+4050+
I need values for +n+ for which +R+%3E=+4050+
+4050+=+-20n%5E2+%2B+135n+%2B+4050+
+0+=+-20n%5E2+%2B+135n+
+0+=+5n%2A%28+-4n+%2B+27+%29+
The solutions are:
+n+=+0+
+-4n+%2B+27+=+0+
+4n+=+27+
+n+=+6.75+
------------------
The price per shoe when +n+=+6.75+ is
+90+-+5n+=+90+-+5%2A6.75+
+90+-+5n+=+90+-+33.75+
+90+-+5n+=+56.25+
----------------------
The range of prices to maintain or increase is
$56.25 to $90
---------------
check:
+R+=+%28+45+%2B+4%2A6.75+%29%2A%28+90+-+5%2A6.75+%29+
+R+=+%28+45+%2B+27+%29%2A56.25+
+R+=+72%2A56.25+
+R+=+4050+
----------------------
(3)
I want the maximum possible +R+
This is where +n%5Bmax%5D+=+-b%2F%282a%29+
+a+=+-20+
+b+=+135+
+n%5Bmax%5D+=+-135+%2F+%28+2%2A%28-20%29%29+
+n%5Bmax%5D+=+3.375+
Plug this back in and find +R%5Bmax%5D+
----------------------------------
+R%5Bmax%5D+=+%28+45+%2B+4%2A3.375+%29%2A%28+90+-+5%2A3.375+%29+
+R%5Bmax%5D+=+%28+45+%2B+13.5+%29%2A%28+90+-+16.875+%29+
+R%5Bmax%5D+=+58.5%2A73.125+
+R%5Bmax%5D+=+4277.81+
The maximum amount by which the store can
increase its revenue from these shoes each week is
$4,277.81
---------------------
Here's the plot of the +R+ function:
+graph%28+500%2C+500%2C+-4%2C+20%2C+-200%2C+4700%2C+-20x%5E2+%2B+135x+%2B+4050+%29+