SOLUTION: Hi, I have a statistics exam on friday and I just have no idea on what to do. Looking at the passed papers this is one of the questions that i just cant figure out. Please help m

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Question 603019: Hi,
I have a statistics exam on friday and I just have no idea on what to do. Looking at the passed papers this is one of the questions that i just cant figure out. Please help me out.
a) A manufacturer of outdoor clothing produces jackets and trousers. Each jacket requires 1 hour to make, whereas each pair of trousers takes 40 minutes. The materials for each jacket cost �32 and those for a pair of trousers cost �40. The manufacturer can devote only 34 hours per week to the production of jackets and trousers and the manufacturer�s total weekly cost for materials must not exceed �1200. The manufacturer sells the jackets at a profit of �12 each and the trousers at a profit of �14 per pair. Market research indicates that the manufacturer can sell all the jackets that are produced.
a) Determine how many jackets and trousers the manufacturer should produce each week in order to maximize profit?
(70 marks)
b)Due to changes in demand, the manufacturer has decided to decrease the number of production hours from 34 hours to 27 hours per week. What effect will this have on the level of profit? Hence estimate the value of the marginal cost of one less hour in production time.
(You may use either an algebraic or graphical approach providing a clear rationale is given to prove that your solution is in fact the optimum.)

Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

Let represent the number of jackets and represent the number of trousers manufactured.

Since we are assuming that you will sell all that you make, the total profit is given by:

and this is the function that we want to maximize subject to the following constraints:

Labor constraint:

Materials constraint:

Non-negative constraints (you can't make a negative number of either jackets or trousers)

Integer constraint (you can't sell fractional parts of jackets or trousers)

Graph each of the constraint inequalities. Note the feasibility polygon that is defined by the region where ALL of the constraint inequalities overlap. A theorem of Linear Programming guarantees that if a unique optimum solution exists, it will be one of the vertices of the feasibility polygon.

Make a list of the ordered pairs that describe all of the vertices of the feasibility polygon. If any of the coordinates of any of the vertices are NOT integers, pick the nearest point to the actual vertex that both has integer coordinates and is inside the feasibility area.

Once you have your table of points with integer coefficients, test each point by substituting the coordinate values in place of and in the objective function (the profit function in this case) and calculate the result. Since this is a maximize problem, select the point that provides the largest solution as your optimum point.

For part b, change the labor constraint inequality to:

then redraw your graph and re-test all of the feasibility vertices. Compare the solution in part a to part b.

John

My calculator said it, I believe it, that settles it