Question

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?

Answer 1

Add and multiply and hopefully you get your answer.

Answer 2

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

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.

`8x+10y\leq 400`

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

`x\geq 15`

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

`profit=55x+80y`

The required linear programming problem is

Maximize profit
`Z=55x+80y`

S.t.,

`8x+10y\leq 400`

`x\geq 15`

`y\geq 0`

Sketch the graph of all inequalities as shown below.

In inequality
`8x+10y\leq 400`, then sign of inequality is ≤. Check the equality by (0,0).

`8(0)+10(0)\leq 400`

`0\leq 400`

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),

`Z=55(15)+80(0)=825`

At (15,28),

`Z=55(15)+80(28)=3065`

At (50,0),

`Z=55(50)+80(0)=2750`

The profit is maximum at (15,28).

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