How do you work out this problem?

Question

A ski manufacturer makes two types of downhill skis. Type A takes 4 labor hours to manufacture and 1 labor hour to finish. Type B takes 3 labor hours to manufacture and 1 labor hour to finish. The manufacturing department has 120 labor hours available per day. The finishing department has
38 labor hours available per day. The profit on type A skis is $65 per pair. The profit on type B skis is $50 per pair. How many pairs of each type should be made each day to maximize profit?

Answer 1

This is a type of problem called "linear programming," where you typically plot a system of inequalities (each of which represents a bound on the production of goods) and then test specific values to figure out how to maximize or minimize a company's profits or costs.

 

STEP 1: Define variables

Here, your variables are number of Type A downhill skis and number of Type B downhill skis:

x = # Type A
y = # Type B

 

STEP 2: Write inequalities that bound production

x ≥ 0 (you can't make a negative number of Type A)
y ≥ 0 (you can't make a negative number of Type B)

4x + 3y ≤ 120 (the company can't use more than 120 hours manufacturing each day; Type A takes 4 hours to manufacture while Type B takes 3 hours)

x + y ≤ 38 (the company can't use more than 38 hours finishing each day; Type A and Type B both take 1 hour to finish)

 

STEP 3: Graph each inequality and find the region of overlap (solution set)

First, change each inequality so it's in slope intercept form. Then, ignore the inequality signs for the moment and just graph the system of inequalities as though they were normal equations:

x = 0 (the y-axis)
y = 0 (the x-axis)
y = (-4/3)x + 40 (a diagonal line with y-intercept 40 and slope -4/3)
y = -x + 38 (a diagonal line with y-intercept 38 and slope -1)

Finally, figure out which region on the graph represents the solution set of the inequalities. You can do this by shading the half of the graph "above" or "below" each inequality based on which half contains solutions to the given inequality. Then, whichever region has been shaded for every single inequality is your solution set. Usually, it's an enclosed polygon shape.

Here, you would shade everything to the right of x = 0 (since that's where x ≥ 0), everything above y = 0 (since that's where y ≥ 0), everything below y = (-4/3)x + 40 (since that's where 4x + 3y ≤ 120), and everything below y = -x + 38 (since that's where x + y ≤ 38).

The solution set for this problem is a quadrilateral bounded by the two axes, y = (-4/3)x + 40, and y = -x + 38.

 

STEP 4: Find points of intersection

Minimizing or maximizing the profit or costs requires you to find the vertices of that solution set's polygon. The maximum or minimum value will always be one of those vertices. To find the vertices, you'll need to know where each pair of lines intersects. Some vertices are straightforward to find, some are not as straightforward.

x = 0 and y = 0:
x and y are already given. The lines intersect at (0, 0).

x = 0 and y = -x + 38:
Plug 0 into the second equation for x to get y = -0 + 38 = 38. The lines intersect at (0, 38).

y = 0 and y = (-4/3)x + 40:
Plug 0 into the second equation for y to get 0 = (-4/3)x + 40, so (4/3)x = 40 and x = 30. The lines intersect at (30, 0).

 

y = (-4/3)x + 40 and y = -x + 38:

Plug (-4/3)x + 40 into the second equation for y to get (-4/3)x + 40 = -x + 38, so 40 = 1/3x + 38 and 2 = 1/3x. That means that x = 6.

Now, plug x = 6 into the second equation to get y = -6 + 38 = 32. The lines intersect at (6, 32). 

 

 

STEP 5: Write an equation for profit and test each vertex for maximization

Now, we'll finally take into account the income from each ski type. Since Type A's make $65 each and Type B's make $50 each, we can represent profit P as:

P = 65x + 50y

Our vertices are (0, 0), (0, 38), (30, 0), and (6, 32). One of these will maximize the profit equation. Plug the values in for x and y to find out which:

(0, 0) yields P = 65(0) + 50(0) = 0 + 0 = $0
(0, 38) yields P = 65(0) + 50(38) = 0 + 1900 = $1,900
(30, 0) yields P = 65(30) + 50(0) = 1950 + 0 = $1,950

(6, 32) yields P = 65(6) + 50(32) = 390 + 1600 = 1990 = $1,990

Since (6, 32) maximizes profit, the company should make 6 Type A skis and 32 Type B skis each day.



You can follow these steps for any linear programming problem you're given. Hope this helps!