An ice cream supplier has two machines that produce vanilla and chocolate ice cream. To meet one of its contractual obligations, the company must produce at least 78 gallons of vanilla ice cream and 22 gallons of chocolate ice cream per day. One machine makes 11 gallons of vanilla and 2 gallons of chocolate ice cream per hour. The second machine makes 6 gallons of vanilla and 5 gallons of chocolate ice cream per hour. It costs $38 per hour to run machine 1 and $38 per hour to run machine 2. How many hours should each machine be operated to fulfill the contract at the least expense?
A)Machine 1 for no hours and machine 2 for 2 hours.
B)Machine 1 for no hours and machine 2 for 6 hours.
C)Machine 1 for 6 hours and machine 2 for 2 hours.
D)Machine 1 for no hours and machine 2 for 13 hours.
E)Machine 1 for 2 hours and machine 2 for no hours.

Question

Quizplus · Accepted Answer

Let x be the number of hours machine 1 should be operated, and y be the number of hours machine 2 should be operated.

The objective is to minimize the cost, which is given by 38x + 38y.

The constraints are:

- 11x + 6y ≥ 78 (minimum gallons of vanilla ice cream)
- 2x + 5y ≥ 22 (minimum gallons of chocolate ice cream)
- x ≥ 0 (non-negative number of hours for machine 1)
- y ≥ 0 (non-negative number of hours for machine 2)

We can solve this problem using linear programming. The feasible region is the intersection of the half-planes defined by the constraints.

The corner points of the feasible region are:

- (0, 4.4) → Cost = 167.2
- (2, 3.33) → Cost = 252.04
- (6, 2.33) → Cost = 341.04
- (7.09, 2) → Cost = 369.82

The minimum cost is achieved at (6, 2.33), which means machine 1 should be operated for 6 hours and machine 2 should be operated for 2 hours. Therefore, the answer is C.

An Ice Cream Supplier Has Two Machines That Produce Vanilla