A travel agency will plan a tour for groups of size 8 or larger. If the group contains exactly 8 people, the price is $800 per person. However, each person's price is reduced by $10 for each additional person above the 8. If the travel agency incurs a price of $100 per person for the tour, what size group will give the agency the maximum profit?

Question

A travel agency will plan a tour for groups of size 8 or larger. If the group contains exactly 8 people, the price is $800 per person. However, each person's price is reduced by $10 for each additional person above the 8. If the travel agency incurs a price of $100 per person for the tour, what size group will give the agency the maximum profit? ​​

Quizplus · Accepted Answer

Let the size of the group be $x$, then the price per person is $800-10(x-8) = 890-10x$.

The revenue the agency receives for a group of size $x$ is therefore $100x(890-10x)$.

Simplifying this expression, we get $R(x) = 89000x-1000x^2$.

The maximum profit will be obtained by choosing the value of $x$ that maximizes $R(x)$.

To find this value of $x$, we take the derivative of $R(x)$ and set it equal to 0:

$R'(x) = 89000-2000x = 0$

Solving for $x$, we get $x=44.5$.

Since the size of the group must be an integer, the best choice would be a group size of 44 or 45. However, since the question asks for the size of the group that gives the MAXIMUM profit, the best answer would be 44, as the revenue would decrease if we choose a larger group size.

Therefore, the best choice would be C) 44.

A Travel Agency Will Plan a Tour for Groups of Size