Question

A shoe manufacturer produces mid-tip and high-top athletics shoes. Cutting machines cut the fabric

for the shoes. Stitching machines stitch the fabric. The cutting machines run 4 hours per day, and the
stitching machines run 5 hours per day. A mid-top shoe takes 1 minute to cut, and 2 minutes to stich. A
high-top shoe takes 3 minutes to cut, and 2 minutes to stitch. The profit is $13 on a mid-top show and
$16 on a high-top shoe. Determine how many shoes of each kind should be made in order to maximize
profits, and what the maximum possible profit is.

Answer 1

Explanation:

Let's use the following variables to represent the number of mid-top and high-top shoes produced per day:

x: number of mid-top shoes

y: number of high-top shoes

We want to maximize the profit function, which is given by:

P(x,y) = 13x + 16y

Subject to the following constraints:

Cutting machines: 1 minute for a mid-top shoe and 3 minutes for a high-top shoe

Stitching machines: 2 minutes for both mid-top and high-top shoes

Cutting machines can run for 4 hours (240 minutes) per day

Stitching machines can run for 5 hours (300 minutes) per day

Non-negative production: x, y >= 0

We can write the constraints in terms of the production time required for each type of shoe:

1x + 3y <= 240 (cutting machines)

2x + 2y <= 300 (stitching machines)

x, y >= 0 (non-negative production)

Now we can graph the feasible region for these constraints and find the corner points:

3y = 240 - x --> y = (240 - x)/3

y = (300 - 2x)/2

x | 0 80 120 180 240

----|-----------------------

y |

----|-----------------------

0 | 0 0 0 0 0

40 | 40 20 0 0 0

150 | 30 60 75 90 100

The profit at each corner point is:

(0, 0): P(0,0) = $0

(80, 40): P(80,40) = $2,480

(120, 0): P(120,0) = $1,560

(150, 0): P(150,0) = $1,950

(180, 0): P(180,0) = $2,340

(240, 0): P(240,0) = $3,120

The maximum profit is achieved at (80,40) where 80 mid-top shoes and 40 high-top shoes are produced per day, with a total profit of $2,480.