A travel agent quotes the flight ticket price for the employees of a company as follows:
When there are 25 people or less, the
ticket price for each person is $1000.
When there are more than 25 people,
with every additional person, the ticket
price for each person will drop by $20.
The lowest ticket price for any employee
is $700.
The company pays $27000 in total. Find the
number of employees going for the trip.

9514 1404 393

  30

Step-by-step explanation:

The total price for x employees will be ...

  p(x) = {1000x for x ≤ 25; x(1000 -20(x -25)) for 25 < x ≤ 40; 700x for 40 < x}

We want p(x) = 27000.

We can check the value of x for the three different functions to see if it is in the specified domain.

  1000x = 27000   ⇒   x = 27 (not in the domain of the first segment)

__

  x(1000 -20(x -25)) = 27000

  20x(75 -x) = 27000 . . . . . collect terms, factor

  (x -37.5)^2 = -1350 +1406.25 . . . . . complete the squre

  x = 37.5 ±√56.25 = 37.5 ±7.5 = {30, 45}

  x =  30 is in the domain; x = 45 is not

__

  700x = 27000

  x ≈ 38.57 (not in the domain of the last segment)

__

So, the piecewise cost function will give a total cost of $27000 when the number of employees going for the trip is 30.