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A hall charges $30 per person for a sports banquet when 120 people attend. For every 10 extra people that attend, the hall will decrease the price by 1.5 per person. What number of people will maximize the revenue for the hall?

Let x = number of groups of extra-ten people
Let R = total revenue

Using this, I created the following quadratic formula:

R = -1.5x^2 + 30x + 3600

Then I complete the square and determine that the vertex is (10, 36150) According to this answer, the number of people which would maximize revenue is 120 + 10 = 130 people; however, my math textbook tells me the answer is 160 people.

What is my mistake? Or is the textbook incorrect?

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2 Answers 2

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if $ x $ number of people attending then you have

$$f(x)=\left\{ \begin{array}{ll} 30x & \mbox{if } x \leq 120 \\ 3600 + {\sum_{i=121}^x({30-\lfloor {\frac{i-120}{10}}\rfloor}1.5}) & \mbox{if } x >120 \\ \end{array} \right.$$

where f is the revenue.

Then you have to differentiate the above function and see where the derivate becomes zero..

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The price per person is $30-1.5x$. The number of people is $120+10x$ . You lost that factor $10$

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