SOLUTION: A fundraising company charges $200 plate for a local charity and sells 300 tickets. For every $10 increase they will lose 5 ticket sales. What would they charge to make a maximum p

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Question 1085633: A fundraising company charges $200 plate for a local charity and sells 300 tickets. For every $10 increase they will lose 5 ticket sales. What would they charge to make a maximum profit and what is the profit?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +n+ = the number of $10
increases in the ticket price
Let +P+ = the profit
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[ profit ] = [ tickets sold ] x [ income from selling 1 ticket ]
+P+=+%28+300+-+5n+%29%2A%28+200+%2B+10n+%29+
+P+=+60000+-+1000n+%2B+3000n+-+50n%5E2+
+P+=+-50n%5E2+%2B+2000n+%2B+60000+
+P+=+-n%5E2+%2B+40n+%2B+1200+
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This is a parabola with a maximum. Use the formula for
the ( n, P ) of the vertex ( maximum in this case )
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+n%5Bmax%5D+=+-b%2F%282a%29+ where
+a+=+-1+
+b+=+40+
+n%5Bmax%5D+=+-40%2F%282%2A%28-1%29%29+
+n%5Bmax%5D+=+20+
So, for a maximum profit, they would charge:
+200+%2B+10n+=+200+%2B+10%2A20+
+200+%2B+10n+=+400+
$400 / plate
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To find the profit, +P+, plug the value for +n+
back into the equation:
+P+=+-n%5E2+%2B+40n+%2B+1200+
+P%5Bmax%5D+=+-20%5E2+%2B+40%2A20+%2B+1200+
+P%5Bmax%5D+=+-400+%2B+800+%2B+1200+
+P%5Bmax%5D+=+1600+
The maximum profit is $1600
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Feel free to get a 2nd opinion if needed