Practice Exercise 4 Solution:

Step 1: Define a variable and create an equation representing the problem. Let x represent the number of students greater than 150 making the school trip.

Largest Gross Income with x = 0 is $2250.

Gross Income with x > 0 is x[15 - .05(150 + x)]

Gross Income = 2250 + x[15 - .05(150 + x)]

GI = 2250 + x(15 – 7.5 - .05x)

GI = 2250 +15x – 7.5x - .05x²

GI = 2250 + 7.5x - .05x²

Step 2: Create a graph of the gross income equation.

The lines below represent GI = 2250 + 7.5x - .05x² dollars and GI = $2531.25.

Step 3: Use the derivative of GI = 0 = 2250 + 7.5x - .05x² to find x where the gross income is greatest.

GI’ = 7.5 - .10x

0 = 7.5 - .10x

-7.5 = -.10x

x = 75

Step 4: Determine the total number of students. Remember, x represents the number of student greater than 150 students.

150 students + 75 students = 225 students

Conclusion: The maximum gross income for the school is when 225 students participate in the trip. Each student will pay $11.25. The maximum school income is $2531.25.