Question 1181585
<pre>
Ticket price as is = $4
Attendence with ticket price as is = 300
Revenue with ticket price as is = (300)($4) = $1200

Let x = the number of ten-cent increases made in the ticket price.
Then the ticket price will increase by $0.10x, making the 
new higher ticket price be $4+$0.10x.

That will also cause the attendance to decrease by 5x.
Then the new lower attendance will be 300-5x.

Let y = the new higher (hopefully) revenue (the amount of money taken in
from selling all the tickets at the new higher ticket price and the new
lower attendance).

{{{y}}}{{{""=""}}}{{{(300-5x)(4+0.10x)}}}

Using FOIL on the right side,

{{{y}}}{{{""=""}}}{{{(300)(4)+(300)(0.10x)+(-5x)(4)+(-5x)(0.10x)}}}

{{{y}}}{{{""=""}}}{{{1200+30x-20x-0.5x^2}}}

{{{y}}}{{{""=""}}}{{{1200+10x-0.5x^2}}}

{{{y}}}{{{""=""}}}{{{-0.5x^2+10x+1200}}}

The maximum new attendance y will occur at the vertex of the parabola,
since the parabola opens downward (since the leading coefficient, a, is
negative).  We use the vertex formula

{{{matrix(1,7,
(matrix(4,1,x-coordinate,of,the,vertex)),
""="",
-b/(2a),
""="",
-(10)/(2(-0.5)),
""="",
10)}}}
 
So we should make 10 ten-cent increases in the ticket price,
which makes the new higher ticket price $5 and the new lower attendance
50 fewer or only 250. So the new higher revenue will be (250)($5) or $1250.

We could also have substituted 10 for x in

{{{y}}}{{{""=""}}}{{{-0.5x^2+10x+1200}}}

and gotten the same answer $1250.

Edwin</pre>