Wahome produces two type of nails - aluminium plated and ordinary nails. It costs 40 cedis to manufacture the ordinary nails and 60 cedis for the aluminium plated. A market research firm states that if an ordinary nail is priced at X cedis and aluminium plated at Y cedis, then Wahome will sell 500(Y - X) of the ordinary nail and 45000 + 500(X - 2Y) of the aluminium plated each year. (a) how should the items be priced to maximize profit? (b) calculate the maximum annual profit of Wahome.

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The cost of producing the estimated quantities of nails is

`40*500*(Y-X)+60*(45000+500(X-2Y)),`

the income is

`X*500*(Y-X)+Y*(45000+500(X-2Y))`

and the profit is the income minus the cost,

`P(X,Y)=(X-40)*500*(Y-X)+(Y-60)*(45000+500(X-2Y))=`

`=500*((X-40)*(Y-X)+(Y-60)*(90+X-2Y)).`

If we open the parentheses, the result will be

`P(X,Y)=500*(-X^2+2XY-2Y^2-20X+170Y-5400).`

 

A maximum of this function is reached or at a point where both partial derivatives are zero, or at the boundaries of the domain. There are some restrictions, at least `Xgt=0` and `Ygt=0.` Also the quantities `500(Y-X)` and `45000+500(X-2Y)` should be nonnegative. This defines a triangle domain with vertices (0, 0), (0, 45) and (90, 90) (check).

 

Let's start from the derivatives:

`(del/(del X)) P(X,Y) = -2X+2Y-20 = 0,` or `-X+Y-10=0,`

`(del/(del Y)) P(X,Y) = 2X-4Y+170 = 0,` or `X-2Y+85=0.`

Adding these equations we obtain `-Y+75=0,` so `Y=75.` Therefore `X=Y-10=65.`

 

Because of the character of `P(X,Y)` (its graph is an elliptic paraboloid vertex up) and the fact that the found point (65, 75) is in the domain (check), we may state that it is the point of maximum profit.

The profit at that point is `P(65, 75) = 500*(25*10+15*(90-85))=500*325=162500` (cedis).

 

The answers: (a) X=65 cedis, Y=75 cedis, (b162500 cedis.
There is also a graph: https://www.desmos.com/calculator/jjr0xhm5vd

 

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