. After retiring, Aunt May decides to start a knitting business. She charges S12

Question

. After retiring, Aunt May decides to start a knitting business. She charges S12 for every knitted item. She spends S10 on knitting equipment and S2 on yarn for each item. Determine her profit function and break-even point write this as a pair). Norman Osborn, the villianous yarn supplier, decides to try keep Aunt May from making too much money. If z is the number of knitted items, then Aunt May now spends 2x per item. What is the maximum profit that Aunt May can make? How can her profits be maximized? Hint: The break-even point question involves a linear equation and the latter question involves a quadratic equation. Both have an initial cost

Explanation / Answer

In the initial condition when aunt may charges 12$ for every knitted item

So let say that she knit x item for breakeven

So the total revenue generated will be 12x $

And the yarn money required for x item will be

2x $

So total profit will be like

12x - 2x

But the initial cost of investement is 10 $ for setting up the business

For breakeven she needs 10$ profit from the business

So when 12x-2x =10 this the breakeven function for aunt may's business

So. When she delivers 1 knitted item she will get the breakeven.

Now for the profit function

Let say aunt may knit x item pe day  

So yarn cost of the day will be like x*2x since norman is charging 2x,per item

So yarn cost will be 2x2

So profit will be P = 12x-2x2

This will be the profit function

P = 12x - 2x2

To get maximize profit per day

Diffenrentiate P w r t x we will get

dP/dx = 12-4x

So to know the maximize value by putting dP/dx =0 (where function is maximum or minimum the derivative of function is zero)

12-4x =0

So x = 3

So at 3 dp/dx =0

So we need to check if at x = 3

d2p/dx2 <0 IF IT IS <0 then the function will give its maximum value at x = 3 ( d2p/dx2 < 0 is the condition for maximizTion of function )

So d2p/dx2 = - 4x so at x=3 it is - 12 which is less than 0

So the function will be maximum at x=3

So for maximum profit aunt may needs to knit 3 items a day

And the profit value will be

P =12x-2x2 at x=3

P = 18 $ of profit per day.

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