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Question

The B & W Leather Company wants to add handmade belts and wallets to its product line. Each belt nets the company $18 in profit, and each wallet nets $12. Both belts and wallets require cutting and sewing. Belts require 2 hours of cutting time and 6 hours of sewing time. Wallets require 3 hours of cutting time and 3 hours of sewing time.

If the cutting machine is available 12 hours a week and the sewing machine is available 18 hours per week, what ratio of belts and wallets will produce the most profit within the constraints?

Answer 1

The first step with word problems like this should always be to write down what we know. Here, we know:

1. b=belts=$18 profit ----> 2 hours cutting, 6 hours sewing
2. w=wallet=$12 profit ----> 3 hours cutting, 3 hours sewing
3. Cutting machine is available 12 hours/week
4. Sewing machine is available 18 hours/week

Now we write equations to define our constraints

Our first constraint is about how many hours a week the cutting machine is available:

2b+3w=12

The second constraint is based on how many hours per week the sewing machine is available:

6b+3w=18

We can use this system of equations to solve for the point where b and w intersect, which will be when the profits are at the highest based on our given constraints. To do this, we will solve for b in our first equation in terms of w.

2b+3w=12

2b=12-3w

b=6-(3w/2)

Now we can plug this b value into our other equation to solve for w!

6b+3w=18

6(6-(3w/2))+3w=18

36-(18w/2)+3w=18

36-9w+3w=18

36-6w=18

-6w=-18

w=3

Now that we know that w=3, we can plug this back into our first equation to solve for b

2b+3w=12

2b+3(3)=12

2b+9=12

2b=3

b=3/2

This means that when our profit is the at the maximum, 1.5 belts are made and 3 wallets are made. This is a ratio of 1.5:3, or 1:2 (as 1.5 is half of 3)

We can make an equation to solve for what the profit will be at this point.

P=18b+12w

P=18(3/2)=12(3)

P=27+36=$63

However, realistically, nobody is going to want to buy 1.5 belts, so this is an example of a question that works well in theory but does not translate well to the real world. If somebody asked you this in the real world, they wouldn't want to know the maximum profit in halves of belts, they would want to know how to split up their time to make full products to maximize their profit. To do this, we are going to test the "floor and ceiling" of each value in the profit equation.

x=1.5, the "floor" of this is 1, the "ceiling" of this is 2

y=3, since this is already an integer, there is no floor or ceiling, it will remain the same. This means that we are going to plug the points (1,3) and (2,3) into our equations to check for the maximum profit that does not violate our constraints (hours the cutting machine and sewing machine are available)

P=18b+12w. (1,3)

P=18(1)+12(3)=18+36= $54

P=18b+12w (2,3)

P=18(2)+12(3)= 36+36= $72 - The profit is higher in this case so we are going to check to make sure that the constraints are not violated if we make 2 belts and 3 wallets.

2b+3w<12

2(2)+3(3)<12

4+9<12

13<12 ------> This does not work, but we are so close!

There is one more option to try.... If rounding up broke a constraint, which it did, we can try expanding our search by decreasing 1 unit of the other variable. This lets us try (2,2)

P=18b+12w

P=18(2)+12(2)=36+24= $60

Now we double check that it matches both constraints!

2b+3w=12

2(2)+3(2)<12

4+6<12

10</=12----> This works!

6b+3w=18

6(2)+3(2)<18

12+6</=18

18</=18---> This works!

The maximum profit using only integers comes when 2 belts and 2 wallets are made!