A Theatre Loses $5$ Seats Every $25$ Cents Added To Ticket Price

Question

So the question is:

A theatre with $1000$ seats can be filled if tickets are $12$ but for every increase in $25$ cents, the theatre will lose $5$ attendees. What is the cost of the ticket that leads to the maximum profit?

I've found this solution and I'm left with the equation below:

Let $x$ be the number of price increases

$y$ is the total profits

$y = -1.25(x-76)^2 + 19220$

Now my question is, why is $76$ the number of price increases? Is there a special rule that makes $(x-76) 0$?

EDIT:

$y = (12+0.25x)(1000-5x)$

$y = 12000-60x+250x-1.25x^2$

$y = -1.25x^2+190x+12000$

$y = -1.25x(x^2-152x)$

$y = -1.25(x^2-152x+5776-5776)+12000$

$y = -1.25(x^2-152x+5776)+7220+12000$

$y = -1.25(x-76)^2+19220$

Answer 1

I haven't checked your result, but let's assume you correctly derived the fact that $$y = -1.25(x-76)^2 + 19220.$$

Now suppose $x$ is not $76$. That is, suppose $x - 76 \neq 0$.

If $x - 76 > 0$, then $(x-76)^2 > 0.$ And if $x - 76 < 0$, then $(x-76)^2 > 0.$ That is, whenever $x - 76 \neq 0$, we end up with $(x-76)^2 > 0.$

When $(x-76)^2 > 0,$ then $-1.25(x-76)^2 < 0.$

When $-1.25(x-76)^2 < 0,$ then $-1.25(x-76)^2 + 19220 < 19220.$

But if $x = 76,$ then $x-76=0$ and $-1.25(x-76)^2 + 19220 = 19220.$

In short, if $x = 76,$ then $y = 19220.$ But if $x$ is anything else, then $y < 19220.$

So the highest possible value of $y$ is $19220$ and the only way to get it is if $x = 76$.

Answer 2

$y = (12+0.25x)(1000-5x)$

$\ \ = 12000+250x-60x-1.25x^2$

$\ \ = 12000+190x-1.25x^2$

Since you have to maximise $y$,

$\ \ \ \ \frac{dy}{dx}=0$

$\Rightarrow190-2.5x=0$

$\Rightarrow x=76$

This is the reason why $76$ shows up.

Answer 3

Consider the standard geometric shifts.

• replacing $x$ in an expression with $x-a$ creates a horizontal shift of $a$ units,

• adding $c$ to an expression creates a vertical shift of $c$ units, and

• multiplying an expression by $k$ scales by $k$

A parabola has exactly one stationary point, and in the most basic form, it is a minimum at the vertex; the negative scaling inverts the parabola and the vertex becomes a maximum.

The basic form can also be described as having a horizontal and vertical shift both of 0, which happens to be the coordinates of the vertex.

If we apply shifts to the parabola, we have in particular shifted the vertex, and since it started at $(0,0)$, it will end up at $(a,c)$.

Thus, the maximum occurs when $x=a$, in your case that is $x=76$ and corresponding value of $y=19220$.

Answer 4

Max[y] = max$[-1.25(x-76)^2+19220)]$

= max[$19220 - 1.25(x – 76)^2]$

= max[$19220$ – a never-negative quantity]

= $19220 - 0$ ;if I have a choice of choosing the value of x; and that choice is when x = 76.