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I am trying to determine a quadratic function to represent the following description.

A travel agenxy offers a group rate of $\$2400$ per person for a week in London if $16$ people sign up for the tour. For each additional person who signs up, the price per person is reduced $\$100$. How many people, in total, must sign up for the tour in order for the travel agency to maximize their revenue? Determine the maximum revenue.

I know to solve the number of people would be to find the axis of symmetry, then for the maximum revenue it would be simply plugging that value into the equation to determine the max revenue.

But how do I set up the quadratic equation??

and I know revenue is equal to (#sold)(cost)

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  • $\begingroup$ When the price becomes $0$, the agency no longer makes money. $\endgroup$ – saulspatz Mar 2 at 18:39
  • $\begingroup$ Well, what goes wrong if you just try to write it down? To be sure, your function is only defined piecewise. If $n≤16$, the function is $f(n)=2400n$. If $n>16$ it is different. To get started, what is $f(17)$? $f(18)$? $\endgroup$ – lulu Mar 2 at 18:39
  • $\begingroup$ would it be the function (2400-100x)(16+x)? where x is the amount of people? $\endgroup$ – fr14 Mar 2 at 18:50
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    $\begingroup$ You have the right idea. The revenue function based on the change in people $x$ can be written as $$R(x)=(2400-100x)(16+x).$$ Using what you know about quadratic functions, you should be able to identify the maximum. $\endgroup$ – Andrew Chin Mar 2 at 18:58
  • $\begingroup$ so then solve for the axis of symmetry to find max people and plug that value into the quadractic equation to determine the max revenue? $\endgroup$ – fr14 Mar 2 at 18:59
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It can be represented by the function:

$(4000-100P)(P)$

where P is the number of people

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  • $\begingroup$ would it be the function (2400-100x)(16+x)? where x is the amount of people? $\endgroup$ – fr14 Mar 2 at 18:51
  • $\begingroup$ No, the function you suggested has a root at $x=24$ and the total cost or even per person cost at $x=24$ isnt zero $\endgroup$ – h-squared Mar 2 at 18:52
  • $\begingroup$ so I cant even make a quadratic function out of this situation? $\endgroup$ – fr14 Mar 2 at 18:53
  • $\begingroup$ Also it cant be represented by a quadratic function since it is defined piecewise before and after $x=16$ $\endgroup$ – h-squared Mar 2 at 18:53
  • $\begingroup$ I think you are interpreting the question incorrectly. $\endgroup$ – Andrew Chin Mar 2 at 18:58

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