You are the operations manager of the Spartan Ice Cream Shoppe that produces its own ice cream. You make and sell five flavors; Vanilla, Chocolate, Pistachio, Banana and Rocky Road. You want to make as much money as you can, of course. The profits per gallon for the five flavors are Vanilla: $90, Chocolate: $75, Pistachio: $70, Banana: $55 and Rocky Road: $40. You can make up to 200 gallons. You have to make at least 10 gallons of each flavor and you cannot make more than 75 gallons of any one flavor. Assume you sell all of the ice cream you make. (a) What is the formula for what you are trying to optimize? (b) What is/are the constraint(s)? (Please answer this question by using the exact sentence mentioned in the question). (c) What is/are the formula(s) for the constraint(s)? (d) What is the solution for this problem? (please use your best logic reasoning)

(a) 90V + 75C + 70P + 55B + 40R

(b) - You can make up to 200 gallons

- You have to make at least 10 gallons of each flavors

- You cannot make more than 75 gallons of any one flavor

(c)V + C + P + B + R ≤ 200

10 ≤ V, C, P, B, R ≤ 75

(d) 75 gallons of Vanilla, 75 gallons of chocolate, 30 gallons of Pistachio, 10 gallons of Banana and 10 gallons of Rocky Road

Step-by-step explanation:

(a)Let V, C, P, B and R be the number of Vanilla, Chocolate, Pistachio, Banana and Rocky Road ice cream gallons you'd plan to make for the Spartan Ice Cream shop. The profit function would be

90V + 75C + 70P + 55B + 40R

And this is the formula that you are trying to optimize.

(b) The constraints are:

- You can make up to 200 gallons

- You have to make at least 10 gallons of each flavors

- You cannot make more than 75 gallons of any one flavor

(c) The constraint formula:

V + C + P + B + R ≤ 200

10 ≤ V, C, P, B, R ≤ 75

(d)  The solutions of this problem is, you would try to make as much as possible of the ones that generate the most profit, while keep an eyes on the constraints. Knowing this, you would

1. Make 10 gallons of each flavors, so a total of 50 gallons. That leaves you with 200 - 50 = 150 gallons left

2. Of the 150 gallons left, you make an extra 65 gallons for Vanilla (so a total of 75 Vanilla gallons). That leaves you with 150 - 65 = 85 gallons

3. Make an extra of another 65 gallons for the 2nd profit generating flavor, which is Chocolate. That leaves you with 85 - 65 = 20 gallons

4. The last 20 gallons can be used to make Pistachio

In the end, you should make: 75 gallons of Vanilla, 75 gallons of chocolate, 30 gallons of Pistachio, 10 gallons of Banana and 10 gallons of Rocky Road

if you was gonna solve for a, a is a   < -12