Question

Aliw theater has seats for 1,000 people. It is filled to capacity for each show and tickets costs PHP 75.00 per show. The organizers went to increase the ticket price. They estimated that for each PHP 25.00 increase in price, 25 fewer people will attend. Write a quadratic function to describe the organizers income after they increase their prize.

A. y = -625 x^2 + 23,125x + 75,000

B. y = 625 x^2 - 23,125x + 75,000

C. y = -625 x^2 + 23,125x - 75,000

D. y = 625 x^2 - 23,125x - 75,000​

Answer 1

Answer:

: A theatre seats 500 people per show and is currently sold out with a ticket price of $16. A survey shows that for every $2 per ticket price increase, 25 fewer tickets will be sold.Write a function to model this situation and use this function to determine the ticket price that will result in the greatest revenue per show.

I tried (25x-500)(2x-16)=0

50x^2-400x-1000x+8000

x^2-28+150

x=-b/a=28/2=$14j

One of the best ways to explain a word problem is by

first looking at an example using an arbitrary number,

like, say, 7. For the reasoning involved in setting

up an equation is the same.

Suppose there are 7 $2 increases.

Then the ticket price is figured by multiplying $2 by 7,

getting $14, and adding to the original price of $16,

getting $14 + 16$ or $30.

Then the number of tickets sold is gotten by multiplying

7 by 25, getting 175 and subtracting from 500, getting

500 - 175, or 325.

Then to get the revenue, we multiply $30 by 325 and get

$9750.

So the equation for the revenue, R(7) is

R(7) = $9750

----------------------

Now let's use that same reasoning, and the same

words, only this time using x instead of 7.

Suppose there are x $2 increases.

Then the ticket price is figured by

multiplying $2 by x, getting $2x, and adding to the

original price of $16, getting $16 + $2x or $2(8 + x).

Then the number of tickets sold is gotten by multiplying

x by 25, getting 25x and subtracting from 500, getting

500 - 25x, or 25(20-x).

Then to get the revenue, we multiply $2(8 + x) by 25(20 - x)

and get $50(8 + x)(20 - x)

So the equation for the revenue, R(x) is

R(x) = $50(8 + x)(20 - x)

Now to get the maximum revenue, we multply

that out

R(x) = 50(160+12x-x²)

R(x) = 8000 + 600x - 50x²

or, in descending order

R(x) = -50x² + 600x + 8000

Now we'll replace R(x) by y and

get the equation

y = -50x² + 600x + 8000