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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?

User Nurchi
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Add and multiply and hopefully you get your answer.
User Matahari
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Answer:

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.

Linear Programming: The polar sports company both manufactures snowboards and skis-example-1
User Fatuhoku
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