Linear programming: the polar sports company both manufactures snowboards and skis. it takes 8 hours to manufacture a snowboard and 10 hours to manufacture a set of skis, there are 400 hours of labor available each week for production. due to demand, they must produce at least 15 snowboards. snowboards profit $55 each and skis profit $80 each. how many should they produce in a week to maximize their profit?

They should produce 15 snowboards and 28 skis in a week to maximize their profit.

Step-by-step explanation:

Let x be the number of snowboards and y be the number of skis.

It takes 8 hours to manufacture a snowboard and 10 hours to manufacture a set of skis, there are 400 hours of labor available each week for production.

Due to demand, they must produce at least 15 snowboards.

Snowboards profit $55 each and skis profit $80 each.

The required linear programming problem is

Maximize profit

S.t.,

Sketch the graph of all inequalities as shown below.

In inequality , then sign of inequality is ≤. Check the equality by (0,0).

This inequality is true for (0,0). It means the shaded region of this inequality is below the line.

The shaded region of x≥15 is right side of vertical line x=15.

The shaded region of y≥0 is above of horizontal line y=0.

From the figure it is clear that the extreme point of common shaded region are (15,0), (15,28) and (50,0).

Find the value of profit function at extreme points.

At (15,0),

At (15,28),

At (50,0),

The profit is maximum at (15,28).

Therefore, they should produce 15 snowboards and 28 skis in a week to maximize their profit.

so firstly, we need to isolate the y variables to be able to solve these inequalities. firstly, add 0.5x on both sides of the first inequality and subtract 1.5x on both sides of the second inequality:

now since the slope is positive for the first inequality, this means that the line going upwards belongs to the first inequality, and the line going downwards belongs to the second inequality.

next, since y ≥ in the first inequality, this means that the shaded region will be above the first inequality's line, thus shading regions a and b.

next, since y ≤ in the second inequality, this means that the shaded region is going to be below this line, thus shading regions a and c.

since both lines shade region a, this means that region a is the solution.

step-by-step explanation:

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