SOLUTION: A video store sells about 100 DVDs a week at a price of $21 each. The owner estimates that for each $3 decrease in price, about 20 more DVDs will be sold each week. How can the ow

Algebra ->  Customizable Word Problem Solvers  -> Finance -> SOLUTION: A video store sells about 100 DVDs a week at a price of $21 each. The owner estimates that for each $3 decrease in price, about 20 more DVDs will be sold each week. How can the ow      Log On

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Question 599430: A video store sells about 100 DVDs a week at a price of $21 each. The owner estimates that for each $3 decrease in price, about 20 more DVDs will be sold each week. How can the owner maximize weekly revenue?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +x+ = the number of $3 decreases in price
Let +R+ = weekly revenue
given:
+R+=+%28+21+-+3x+%29%2A%28+100+%2B+20x+%29+
+R+=+2100+-+300x+%2B+420x+-+60x%5E2+
+R+=+-60x%5E2+%2B+120x+%2B+2100+
+R%2F60+=+-x%5E2+%2B+2x+%2B+35+
This parabola has a maximum since the
coefficient of the x%5E2 term is minus
The maximum is at x+=++-b%2F%282a%29+ when
the equation has the form
+ax%5E2+%2B+b%2Ax+%2B+c+
+a+=+-1+
+b+=+2+
+-b%2F%282a%29+=+-2%2F%282%2A%28-1%29%29+
+-b%2F%282a%29+=+1+
When +x+=+1+,
+R%2F60+=+-%281%29%5E2+%2B+2%2A1+%2B+35+
+R%2F60+=+-1+%2B+2+%2B+35+
+R%2F60+=+36+
+R+=+2160+
The owner can maximize revenue by
decreasing the price once by $3/CD
check:
+x+=+0+
+R+=+%28+21+-+3%2A0+%29%2A%28+100+%2B+20%2A0+%29+
+R+=+2100+
-------------
+x+=+1+
+R+=+%28+21+-+3%2A1+%29%2A%28+100+%2B+20%2A1+%29+
+R+=+18%2A120+
+R+=+2160+
-------------
+x+=+2+
+R+=+%28+21+-+3%2A2+%29%2A%28+100+%2B+20%2A2+%29+
+R+=+15%2A140+
+R+=+2100+
OK