SOLUTION: The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 63 students, requires 2 chaperones, and costs $1200 to rent.

Question 1097065: The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 63 students, requires 2 chaperones, and costs $1200 to rent. Each van can transport 7 students, requires 1 chaperone, and costs $120 to rent. Since there are 252 students in the senior class that may be eligible to go on the trip, the officers must plan to accommodate at least 252 students. Since only 22 parents have volunteered to serve as chaperones, the officers must plan to use at most 22 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation; costs? What are the minimal transportation costs?
Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website!
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The restrictions are

  63*B + 7*V >= 252           (1)    (to transport all 252 students)
   2*B + 1*V <=  22           (2)    (chaperones)

     B >= 0,                  (3)
     V >= 0.                  (4)

The objective function to minimize is  

  F(B,V) = 1200*B + 120*V.    (5)    (transportation cost)


The feasible domain is shown in the figure below

It is the triangle domain in the Quadrant I restricted by the straight lines (inequalities)

  63*B + 7*V >= 252           (1)   (red line)
   2*B + 1*V <=  22           (2)   (green line)

     V >= 0                   (4)


So, it is the area ABOVE the red sloped line and BELOW the green line






lines  63*B + 7*V = 252 (red)  and  2*B + 1*V = 22 (green)


The critical points are 

    P1 = (2,18)  (intersection of lines 1) and (2))       F(2,18) = 1200*2 + 120*18 = 4560.

    P2 = (4,0)   (B-intercept for the red line   (1))     F(4,0) = 1200*4 + 120*0 = 4800.

    P3 = (11,0)  (B-intercept for the green line (2))     F(0,22) = 1200*11 + 120*0 = 13200.


Calculate the function F(B,V) in each of the points P1, P2 and P3.

Then select the point where the function F(B,V) is minimal.

The Linear Programming Method states that this point is the solution: it provides minimal transportation cost.


So, in our case the optimal solution is 2 buses and 18 vans.  
The transportation cost is 4560 dollars in this case.

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