SOLUTION: A baseball team plays in a stadium that holds 68000 spectators. With the ticket price at $11 the average attendance has been 28000. When the price dropped to $9, the average attend

Algebra ->  Finance -> SOLUTION: A baseball team plays in a stadium that holds 68000 spectators. With the ticket price at $11 the average attendance has been 28000. When the price dropped to $9, the average attend      Log On


   

Question 1136931: A baseball team plays in a stadium that holds 68000 spectators. With the ticket price at $11 the average attendance has been 28000. When the price dropped to $9, the average attendance rose to 34000. Assume that attendance is linearly related to ticket price.
What ticket price would maximize revenue?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +y+ = attendance in thousands
Let +x+ = ticket price in dollars
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You are given 2 points on a straight line:
( 11, 28 )
( 9, 34 )
Use point-slope formula
+%28+y+-+28+%29+%2F+%28+x+-+11+%29+=+%28+34+-+28+%29+%2F+%28+9+-+11+%29+
+%28+y+-+28+%29+%2F+%28+x+-+11+%29+=+6%2F%28-2%29+
++%28+y+-+28+%29+%2F+%28+x+-+11+%29+=+-3+
+y+-+28+=+-3%2A%28+x+-+11+%29+
+y+-+28+=+-3x+%2B+33+
+y+=+-3x+%2B+61+
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check:
( 11,28 )
+28+=+-3%2A11+%2B+61+
+28+=+-33+%2B+61+
+28+=+28+
OK
( 9,34 )
+34+=+-3%2A9+%2B+61+
+34+=+-27+%2B+61+
+34+=+34+
OK
------------------------
Revenue, R = [ ticket price ] x [ attendance ]
+R+=+x%2A%28+-3x+%2B+61+%29+
+R+=+-3x%5E2+%2B+61x+
+x%5Bmax%5D+=+-b%2F%282a%29+
+x%5Bmax%5D+=+-61%2F%28+2%2A%28-3%29%29+
+x%5Bmax%5D+=+61%2F6+%7D%7D%0D%0A%7B%7B%7B+x%5Bmax%5D+=+10.17+
------------------------------
+R%5Bmax%5D+=+%28+61%2F6+%29%2A%28+-3%2A%2861%2F6%29+%2B+61+%29+
+R%5Bmax%5D+=+10.1667%2A%28+-30.5+%2B+61+%29+
+R%5Bmax%5D+=+10.1667%2A30.5+
+R%5Bmax%5D+=+310.0843+
$310,084
If you need to, get a 2nd opinion also
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Here's the plots of +y+ and +R+
+graph%28+400%2C+400%2C+-5%2C+30%2C+-10%2C+70%2C+-3x+%2B+61++%29+
+graph%28+400%2C+400%2C+-5%2C+30%2C+-40%2C+400%2C+-3x%5E2+%2B+61x+%29+