A small candy shop is preparing for the holiday season. The owner must decide ho
ID: 408259 • Letter: A
Question
A small candy shop is preparing for the holiday season. The owner must decide how many bags of deluxe mix and how many bags of standard mix of Peanut/Raisin Delete to put up. The deluxe mix has 2/3 pound raisins and 1/3 pound peanuts, and the standard mix has 1/2 pound raisins and 1/2 pound peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with. Peanuts cost $.60 per pound and raisins cost $1.0 per pound. The deluxe mix will sell for $2.90 per pound, and the standard mix will sell for $2.55 per pound. The owner estimates that no more than 110 bags of one type can be sold If the goal is to maximise profits, how many bags of each type should be prepared? What is the expected profit?Explanation / Answer
Answer:
Let x represent bags of deluxe mix
and y represent bags of standard mix.
The profit on the one-pound bag of deluxe mix is
$2.34 - $1.50(2/3) - $0.60(1/3) = $2.34 - $1 - $0.20 = $1.14
The profit on the one-pound bag of standard mix is
$2.51 - $1.50(1/2) - $0.60(1/2) = $2.51 - $0.75 - $0.30 = $1.46
[Query here: Did the prices get switched around in this question? Deluxe mix sells for less than standard, although its ingredients are more expensive? If so, the calculation for maximizing the profit function will have to be redone. But I'll go ahead and use the prices listed.]
Constraints:
x >= 0
y >= 0
x <= 110
y <= 110
(2/3)x + (1/2)y <= 71 [pounds of raisins available]
(1/3)x + (1/2)y <= 57 [pounds of peanuts]
Multiplying those last two constraints by 6 gets rid of the fractions:
4x + 3y <= 426
2x + 3y <= 342
These two constraints's borderlines intersect at (42,86).
To the left of that, 4x+3y<=426 is the constraint in force;
it intersects the y-axis at (106.5,0).
To the right of (42,86), 2x+3y<=342 is in force;
it intersects y=110 at (6,110).
The other vertices of the feasible area are (0,0) and (0,110), and they clearly cannot maximize profit.
If the prices on the mixes are as stated, the profit function is
p(x,y) = $1,14x + $1.46y and the profit at each vertex is
(106.5,0): $121.41
(42,86): $173.44
(6,110): $167.44
so 42 bags of deluxe mix and 86 bags of standard mix should be prepared,
for a profit of $173.44.