Question

During the summer months terry makes necklaces to sell on the beach. Last summer he sold necklaces for 10 dollars each and averaged 20 sales per day. When he increased the price by 1 dollar he found that he lost 2 sales per day. If the material for each necklace costs Terry 6 dollars what should the selling prices be to maximize his profit? Profit=Revenue-Cost

Answer 1

gotta be over 6 bucks thats for sure

Answer 2

figured that :)

Answer 3

make a linear graph; they give you two points for reference

10,20 and 11,18

Answer 4

what would the linear equation look like?

Answer 5

this is your demand curve then

as x moves by 1; y moves by -2; so its a slope of -2

-2x +2(10)+20

-2x +20 +20

D(x) = -2x +40 ; right?

Answer 6

so, then i would take the derivative of that?

Answer 7

the supply curve is simply 6x

Answer 8

where the supply and demand curve meet; should be the selling price

Answer 9

y = -2x +40
y = 6x

6x = -2x +40

8x = 40; x = 5 might work out

Answer 10

can you show me how to do this using optimization

Answer 11

dunno; since its all linear; derivatives are kinda pointless .... but there might be something im overlooking in general

Answer 12

-2(5)+40 = -10+40 = 30 bucks made

6(5) = 30 spent; so that more of a breakeven point

Answer 13

x = price in one and quantity in the other .... thats one error

Answer 14

number sold = (-2x+40)

Revenue = (-2x+40)*x

Costs = 6(number sold)
= 6(-2x+40)


Profits = revenue - costs

P(x) = -2x^2 +40x +12x -240
= -2x^2 +62x -240 .......... perhaps?

Answer 15

then take the derivative of that

Answer 16

sure, cant hurt :)

Answer 17

but do you think that would be a good idea to do, then i would find the critical values

Answer 18

if the equation is good, and thats a big if these days, then yes, the derivative will help you find where the profits peak at

Answer 19

I get a price of $15.50 lets see if it works

-2(15.50) +40 = 9 sold

9(15.50) = 139.50 in revenue

6(9) = 45 in costs

P(x) = 94.50
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how does that go for 15 and 16?

15: #sold = 10; revenue = 150, costs = 60, profit = 90

Answer 20

16: #sold = 8, revenue = 128; costs = 48, profit = 80

Answer 21

you got 15.50 from taking the derivative and finding the critical point right

Answer 22

yes:

the derivative of the profit: -4x +62

-4x +62 = 0

-4x = -62

x = -62/-4 = 15.50

Answer 23

and you decided to test 16 just to check right