A student makes and sells necklaces at the beach during the summer months. The material for each necklace costs her $6 and she has been selling about 24 per day at $10 each. She has been wondering whether or not to raise the price, so she takes a survey and finds that for every dollar increase she would lose 2 sales a day.
(a) If she increases her price by x dollars, then her price will be dollars, her number of sales for the day will be necklaces, and her cost to produce the necklaces for a day will be dollars.
(b) Write her profit P as a function of the number x of dollar increases in price:
(b) What price (to the nearest cent) should she set for the necklaces to maximize her profit?
(c) What is the maximum profit (to the nearest cent)?
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Expert Chegg Question Answer:
a) As her current price is $10, if she increases the price by $x, it will be $(10+x)
sales go down at 2 per $1 increase, so sales will be 24 – 2x
Material cost is 6(# of sales) = 6(24-2x)
b) revenue = (10+x)(24-2x)
cost = 6(24-2x)
profit = (10+x-6)(24-2x) = (4+x)(24-2x) = 96 + 16x -2x2
Completing the square, -2(x2 – 8x) + 96 =
-2(x2 – 8x + 16) + 32 + 96 =
-2(x-4)2 + 128
The maximum occurs when x-4 = 0, or x = 4
Then, the price is 10+ x = 10 + 4 = $14
c) The maximum profit is $128 (the squared term is 0 and we only have the constant term)
As a check, (10+4)(24-2*4) – 6(24-2*4) = 8(16) = 128