Chapter 4.4, Problem 27E

To determine

To find: The number of people that is required to maximize the profit of a tour service.

Answer to Problem 27E

The required number of people is 67.

Explanation of Solution

Given information:

1. $200 per person if 50 people go on tour.
2. For each extra person, up to at most 80 people, the rate per person is decreased by $2.

Also, the fixed cost is $6000 plus $32 per person who go to the tour.

Concept used: The maximum value of the function $f\left(x\right)$ will be at the critical point of that function.

Calculation:

From the given statement, write the revenue function $r\left(x\right)$ and the cost function.

  $c\left(x\right)$ .

  $r\left(x\right)=\left[200-2\left(x-50\right)\right]x$

  $c\left(x\right)=6000+32x$

Subtract the cost function from the revenue function to obtain the profit function, $p\left(x\right)$ .

  $\begin{array}{c}p\left(x\right)=r\left(x\right)-c\left(x\right)\\ =\left[200-2\left(x-50\right)\right]x-6000-32x\\ =200x-2x^2+100x-6000-32x\\ =-2x^2+268x-6000\end{array}$

Differentiate the obtained function with respect to $x$ .

  $p^{\prime }\left(x\right)=-4x+268$

Equate the obtained derivative to 0 and solve for $x$ .

  $\begin{array}{c}{p}^{\prime }\left(x\right)=0\\ -4x+268=0\\ x=67\end{array}$

Now, differentiate the equation, $p^{\prime }\left(x\right)=-4x+268$ , with respect to $x$ .

  $p^{″}\left(x\right)=-4$

Since at $x=67$ , $p^{″}\left(x\right)<0$ , the profit will be maximum at $x=67$ .

So, the required number of people is 67.