. 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

Given that, the initial fixed cost of equipment = $10

Per $2 on yarn of each item.

Total no.of knitted items = x

So, total cost for yarn of knitted items = 2x.

So, total cost function = 10+2x

Revenue per 1 item = $12

Revenue per x items = 12x

Profit = revenue-cost

Profit function P(x) = 12x-10-2x

=10x-10

Break even point is where no profit, no loss.i.e profit = 0

10x-10 = 0

x =1

Aunt May has to knit 1 item atleast to be in break even point.

b)

In this case, given that for x knitted items, each item costs 2x along with the fixed inital cost of $10 for equipment.

So, Cost function=

Fixed cost+knitting yarn cost

= 10+2x2

Revenue = 12x for x items

Profit = revenue-cost

= 12x-10-2x2

We need to find the maximum profit in this case,

for a quadratic equation, ax2+bx+c -=0, maximum occurs at x = -b/2a

Similarly here, x =-12/-4 =3

So, at x = 3 her profit is maximized.substituting the value in the profit function give us,

maximum profit = 36-28 = $8

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