Question 564173: Hi, I am having a problem forming the equation from a worded problem. I am struggling to understand what the terms mean in this equation and how they worked them out. I understand how to solve and can write quadratic motion equations. I have a good understanding.
The problem is: A school concert usually attracts 600 people at a cost of $10 per person. On avaerage for every $1 rise in admission price, 50 less people attend the concert. If T is the total amount of takings and n is the number of $1 increases write a rule for the function which gives T in terms of n.
The answer is: T = 6000 + 100n -50n^2
My thinking is that the 6000 represents the miinmum of the independant variable 600 people at $10 each which totals $6000. So this is where the parabola will start from the y-axis (i.e at zero). But how did they get the 100n? What does this mean in the context of the problem and why n^2?
Why isn't it linear? Like T = 6000 - 50(n)
Your help would be appreciated.
Found 2 solutions by ankor@dixie-net.com, bucky:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! A school concert usually attracts 600 people at a cost of $10 per person.
On average for every $1 rise in admission price, 50 less people attend the concert.
If T is the total amount of takings and n is the number of $1 increases write a rule for the function which gives T in terms of n.
The equation:
Total = no. of people * cost for each ticket
Which is
T(n) = (600-50n)*(10+n)
FOIL
T(n) = 6000 + 600n - 500n - 50n^2
The quadratic equation which has max value for some value of n
T(n) = -50n^2 + 100n + 6000
:
The parabola will look like this
Looks like max revenue will occur when n=1, 550 people and $11 ticket $6050
Answer by bucky(2189)  (Show Source):
You can put this solution on YOUR website! You were close to getting the process when you correctly determined that the $6000 was determined by multiplying the 600 attendees by 10 dollars for a ticket.
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The basic question to ask yourself is, "How is the total Take from the concert calculated?" The answer is that it is determined by multiplying the number of tickets sold times the cost per ticket.
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So first you need to determine the number of tickets that will be purchased. We start with 600 and take away 50 for each $1 increase in the price of the tickets. Letting n represent the number of $1 increases, we can say that the number of tickets that will be sold is given by:
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Tickets sold = 600 - 50*n
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(Again you were pretty close to that when you proposed a linear solution.}
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At the same time, the price per ticket is going up $1 for each increase of 1 in the value of n. So we can say:
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Price per ticket = 10 + n
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Now we can determine the Take (T) by multiplying these two factors (tickets sold times price per ticket) as follows:
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T = (600 - 50*n)*(10 + n)
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Multiplying out the right side using the FOIL process (this first involves multiplying 600 times each of the terms in the second set of parentheses and then multiplying -50*n times each of the terms in the second set of parentheses) we get:
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T = 6000 + 600*n - 500*n - 50*n^2
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Next we combine the two terms in the middle (both having n) to get the answer of:
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T = 6000 + 100*n - 50*n^2
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and there's the answer we were trying to find.
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One of the things that we also should probably do is to specify a range of limits on n. For example, we should probably limit the minimum value of n to 0 because if we let n be negative we would be decreasing the price per ticket below $10, not increasing it above $10 as the problem proposes. Note that the problem gives no indication that the number of tickets purchased when the price goes down is likely to increase by 50 persons for each $1 decrease in the price of a ticket. This means that we don't know what to expect will happen to the attendance for each $1 decrease in the ticket price. Next we should probably say that n cannot exceed +12. Why? Because if it did the number of tickets sold (600 - 50*12) would likely be zero (600 - 600). (Selling a negative number of tickets would not make sense.)
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Now just for a "fun" exercise, let's make a graph of the Take (vertical axis) relative to the value of n (horizontal axis). Here it is:
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We have already said that the portion of the graph that is of interest in this problem is in the first quadrant because our rough analysis said that n should be in the range of zero to positive 12. (Notice that the marks along the horizontal axis are in units of 2, 4, 6, etc.)
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The graph adds a little more detail. From the graph we can see that the maximum value of n (along the positive horizontal axis) should be 12 or less so that the Take (along the vertical axis) is not negative. (This confirms the rough analysis that we did earlier.)
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Notice something else of interest. The peak of graph does not appear to occur where n is zero (meaning where the price of a ticket is $10). In fact, from the graph we can determine that the intercepts on the horizontal axis occur at -10 and +12. The peak will occur midway between these two values, and we can determine that value by averaging -10 and +12 as follows:
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Average n = (-10 + 12)/2 = 2/2 = 1
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Therefore, we can predict that the peak of the graph (that is, the maximum value of the Take) will occur when n = 1.
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As confirmation, if we use 1 as the value of n and substitute that into the equation for T, we find that for this $1 increase in the price of a ticket the Take will be $6050, even though the number of persons buying a ticket is reduced by 50 (meaning an attendance of 550 persons). Interesting development. Maybe the school should really be selling $11 tickets (meaning n = 1) and expecting 550 tickets to be sold. The intake from the concert would likely increase by $50 (from $6000 when 600 tickets are sold at $10 each to $6050 when 550 tickets are sold at $11 each), not to mention the possibilities that the school might save some more dollars by printing less tickets and less concert programs. However, maybe the real object of the concert is not to maximize the Take, but rather to cover the costs of producing and performing the concert and after those costs are covered the real aim is to expose the largest possible audience to the arts. Just some "fun" things to think about.
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Hope this helps you to understand the problem a little better.
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