SOLUTION: a travel agency offers an organization an all-inclusive tour for $800 per person if nor more than 100 people take the tour. however the cost per person will be reduced to $5 for ea
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Question 1095701: a travel agency offers an organization an all-inclusive tour for $800 per person if nor more than 100 people take the tour. however the cost per person will be reduced to $5 for each person in excess of 100. how many people should take the tour in order for the agency to receive the largest gross revenue, and what is the largest gross revenue?
The condition says that in the domain n >= 100 the following formula works for the single ticket price P:
P = 800 - 5*(n-100) dollars.
The revenue R is the product R = n*P, or
R = n*(800-5*(n-100)) = 800n - 5n^2 + 500n = 1300n - 5n^2 = 5n*(260-n).
The function R(x) = 5x*(260-x) is the parabola, and you need to find its maximum.
This parabola has the roots x = 0 and x = 260.
Therefore, its maximum is exactly mid-point between the roots, i.e. x = 130.
Thus, the optimal number of people in the tour is 130: this number gives the maximum to Revenue.
To calculate the value of this maximum, substitute x = n = 130 into the parabola equation
R(n) = 5n*(260-n) = 5*130*130 and calculate.
