plz i need help in a math word problems?

2.A baseball team plays in a stadium that holds 45,000 spectators. With the ticket price at $18, the average attendance at recent games has been 22,000. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 2,000.
a.Find a function that models the revenue in terms of the ticket price.
b.What ticket price is so high that no revenue is generated? Use this to determine a reasonable domain for the revenue function.
c.Find the price that maximizes revenue from ticket sales.

Let the ticket price be t.

When t=18, attendance=22,000
When t=18-1=17,
attendance=22,000+2,000=24,000
When t=17-1=16,
attendance=24,000+2,000=26,000
When t=18-x, attendance=22,000+2,000x
In terms of t, x=18-t, so:
attendance=22,000+2000(18-t)
=22,000+36,000-2000t=58,000-2000t

Since revenue=ticket price*average attendance

Then,

r(t)=t*(58000-2000t)
r(t)=2000t(29-t)
You may want to write this as:
r(t)=58,000t-2000t^2

“What ticket price is so high that no revenue is generated?”
The key is “no revenue.” Thus, r(t)=0.

0=58,000t-2000t^2
Clearly t=0, but this is a bit obvious. The second would have to be:
0=58,000-2000t (by dividing by the known factor of t)
58,000=2000t
58,000/2,000=t
29=t

When the ticket price is $29, there will be no revenue generated.

“Find the price that maximizes revenue.”

Taking the first derivative,

r’(t)=58,000-4000t

r’(t)=0 at the vertex

0=58,000-4000t at the vertex
58,000=4000t at the vertex
t=14.5 at the vertex

When the price is $14.50, we generate the highest revenue.

Note: You may not want to use derivatives. In this case, find the average of the two roots. Since the roots were t=29 and t=0, then the vertex would have to t={29+0}/2=29/2=14.5

Hope this helped. Peace

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