Question

Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?

Answer 1

Grade A = 45 lbs Grade B = 70 lbs Total weight = A+B = 45+70 = 115 Blend Br = 1/3A + 2/3B Blend Af = 1/2A + 1/2B Profit = 1.50*lbs Blend Br + 2.00*lbs Blend Af Profit = 1.50*(1/3A + 2/3B)+2,00(1/2A + 1/2B) = 45 pounds of A and 70 pounds of B yields max 24A + 72B = 98 Pounds Max Br " " " " max 38A + 76B = 114 Pounds Max Af The rate of profit for A is .50 per pound in BR and 1.00 per Pound in AF For B, rate of profit for B is 1.00 for Br and 1.00 for Af. Let X = # pounds in A Let Y = # pounds in B 115 = A + B A = 115-B Let q = percent of Br Let r = percent of Af Let s = pounds of A in Br then 45-s = # pounds A in Af Let t = pounds of B in Br and 70-t= # pounds in Af .5s + 1.00*(45-s) + 1.00(t) + 1.00(70-t) = p 1.5s + 2.00t = p If s = 45 pounds then 70 = 70/45 = 14/9s pounds .5s + 1(45-s) + 1(14/9s) + 1(70-14/9s) = p .5*45s + (45(1 - s)) + 14/9*45s + (45(1 - 14/9s) =p 22.5s + 45 - 45s + 70s + 35 - 70s = p 22.5s +70 - 115s = p p = 137.5s + 70 p' = 137.5 22.5 A and 45 B for Br and 23.75 A and 23.75 B for Af. Max profit is $137.5

Answer 2

105 pounds of breakfast and 90 pounds of afternoon packages

profit = $337.5

Explanation:

Defining:

Xb: number of 1 pound breakfast packages produced

Xa: number of 1 pound afternoon packages produced

The optimization problem is the following:

Maximize profit = Xb * $1.5 + Xa * $2

Subject to

Xb*1/3 <= 45

Xb*2/3 <= 70

Xa*1/2 <= 45

Xa*1/2 <= 70

45 is the total amount of A grade tea; 70 is the total amount of B grade tea;

In the figure we can see feasible region (every point in this zone fulfill restrictions). We know that optimal point will be in the corner of the feasible region, in our case points are (0,0) (0,90) (105, 90) (105, 0). Given that we want to maximize, it is obvious that the answer is Xb = 105 and Xa = 90. Replacing in profit's equation: 105 * $1.5 + 90 * $2 = $337.5