```Question 131503
You are an event coordinator for a large performance theater. One of the hottest new Broadway musicals has started to tour and your city is the first stop on the tour. You need to supply information about projected ticket sales to the box office manager. You provide the manager with a quadratic equation that models the expected number of ticket sales for each day x. (x = 1 is the day tickets go on sale).
Tickets = {{{-0.2x^2 + 12x + 11}}}
<pre><font size = 4 color = "indigo"><b>
Here's the graph (only the even numbers are marked on the axes
and the odd numbers are in between them.

{{{graph(801,2201,-10,70,-10,210,-0.2x^2 + 12x + 11)}}}
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a. Does the graph of this equation open up or down?
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Down.
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<pre><font size = 4 color = "indigo"><b>
By the fact that the coefficient of {{{x^2}}} is negative
</pre></font></b>
b. Describe what happens to the tickets sales as time passes?
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They increase from about 23 the first day up to about 191 on the
30th day, then they decrease down to nothing by about the 61st day.
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c. Use the quadratic equation to determine the last day that tickets will be sold. (Note: Write your answer in terms of the number of days after ticket sales begin.)
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The last day will be when the number of tickets is nearest to zero.  So
we set the expression for the tickets = 0 to find the day when the number
of tickets is nearest zero.

{{{-0.2x^2 + 12x + 11 = 0}}}

Tickets = {{{(-b +- sqrt( b^2-4*a*c ))/(2*a) }}}

where a = -0.2, b = 12, c = 11

Tickets = {{{(-12 +- sqrt( 12^2-4*-0.2*11 ))/(2*-0.2) }}}

Tickets = {{{(-12 +- sqrt( 144+8.8 ))/(-0.4) }}}

Tickets = {{{(-12 +- sqrt( 152.8 ))/(-0.4) }}}

Tickets = {{{(-12 +- 12.36122971)/(-0.4) }}}

Tickets = {{{(-12 +- 12.36122971)/(-0.4) }}}

Tickets = {{{(-12 +- 12.36122971)/(-0.4) }}}

Using the +

Tickets = {{{(-12 + 12.36122971)/(-0.4) }}}

Tickets = {{{(0.3612297123)/(-0.4) }}}

Tickets = {{{-.9030742807}}}

That won't do since it's negative.

Using the -

Tickets = {{{(-12 - 12.36122971)/(-0.4) }}}

Tickets = {{{(-24.36122971)/(-0.4) }}}

Tickets = {{{60.90307428}}}

That answer is OK. That means the last time tickets
will be sold is on the 60th day.
be sold.
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d. Will ticket peak or be at a low during the middle of the sale?
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peak
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How do you know?
<pre><font size = 4 color = "indigo"><b>
By looking at the graph at the 30th day, which is right in the
middle of the sale.
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e. After how many days will the peak or low occur?
<pre><font size = 4 color = "indigo"><b>
The peak will occur on the 30th day and the low looks like
it is the 60th day.
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f. How many tickets will be sold on the day when the peak or low occurs?
<pre><font size = 4 color = "indigo"><b>
The peak will occur on the 30th day.  To determine how many tickets will
be sold on the 30th day, we substitute x = 30

Tickets = {{{-0.2x^2 + 12x + 11}}}
Tickets = {{{-0.2(30)^2 + 12(30) + 11}}}
Tickets = {{{-0.2(900) + 360 + 11}}}
Tickets = {{{-180 + 360 + 11}}}
Tickets = {{{191}}}

The low day is when x = 60

To find out how many tickets are sold when x = 60 (the last day)

Tickets = {{{-0.2x^2 + 12x + 11}}}
Tickets = {{{-0.2(60)^2 + 12(60) + 11}}}
Tickets = {{{-0.2(60) + 720 + 11}}}
Tickets = {{{-720 + 720 + 11}}}
Tickets = {{{11}}}
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g. What is the point of the vertex?
<pre><font size = 4 color = "indigo"><b>
The vertex is the point (30,191)

We found that with the vertex formula:

Vertex = ({{{-b/(2a)}}},{{{-d/(4a)}}})

where d = discriminant {{{d=b^2-4ac}}} =

{{{d=b^2-4ac}}} = what was under the radical
in the quadratic formula or 152.8

So the vertex is

Vertex = ({{{-b/(2a)}}},{{{-d/(4a)}}})

Vertex = ({{{-12/(2(-0.2))}}},{{{-152.8/(4(-0.2))}}})

Vertex = ({{{-12/(-0.4))}}},{{{-152.8/(-0.8))}}})

Vertex = ({{{30}}},{{{191}}})
</pre></font></b>
How does the number related to your answers in parts e and f?
<pre><font size = 4 color = "indigo"><b>
The jibe very nicely with the peak point determined by
looking at the graph.
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h. How many solutions are there to the equation -0.2x^2 + 12x + 11 = 0?
<pre><font size = 4 color = "indigo"><b>
Two
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How do you know?
<pre><font size = 4 color = "indigo"><b>
I found them both using the quadratic formula in part (c).
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i. what do the solutions represent?
<pre><font size = 4 color = "indigo"><b>
The x-intercepts of the graphs.
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Is there a solution that does not make sense?
<pre><font size = 4 color = "indigo"><b>
Yes, the solution "Tickets = {{{-.9030742807}}}"
</pre></font></b>
If so, in what ways does the solution not make sense?
<pre><font size = 4 color = "indigo"><b>
The number of tickets cannot be negative.

Edwin</pre>```