Question

Really hard Calculus problem. Anyone up to the challenge?

Answer 1

During the summer months, Terry makes and sells necklaces on the beach. Last
summer, he sold the necklaces for $10 each and his sales averaged 20 per day. He also found that for each $1 increase in price sales drop by two per day. If the material for each necklace costs Terry $6, what should the selling price be to maximize his profit?

Answer 2

Ok, I think I got it.

So let \(s\) be sales and \(p\) be price.

We have \(Profit = ps - 6s\), right?

Answer 3

And we need to find relationship between p and s.

Answer 4

profit is revenue-cost

Answer 5

What do you mean?

Answer 6

that what profit maximization is

Answer 7

ok, so you disagree with what I said?

Answer 8

Yes

Answer 9

why?

Answer 10

Because on the answer sheet its like this: (10-x)(20-2x)-6(20-2x)

Answer 11

The left side before the subtraction is the revenue and the right side of the subtraction is the cost

Answer 12

maybe this will make more sense http://www.math.rutgers.edu/~sferry/MA135F14/135-f14/Final2008.pdf
Number 10

Answer 13

What does x represent?

Answer 14

The amount of items he must sell

Answer 15

and I got $13 my way

Answer 16

How?

Answer 17

okay, give me a min how to merge it your way

Answer 18

Dont do my way

Answer 19

If there is a shortcut tell me

Answer 20

You can draw it out if it makes it easier

Answer 21

The demand function is the price p(x) per necklace Terry can charge and if he sells x necklaces. If this is a linear function then

p(x) = ax + b where a and b are constants.

Last summer when the price was $10 he sold 20 necklaces per day so

$10 = p(20) = 20 a + b

When he increased the price to $10 + $1 = $11 he lost 2 sales per day. Thus

$11 = p(18) = 18 a + b

Solve these equations for a and b.

The revenue that Terry makes is R(x)= x * p(x) and his cost is C(x) = 6 x Write an expression for the profit he makes and use your knowledge of calculus to maximize the profit

Answer 22

I would like to see another way also

Answer 23

Thanks that is useful

Answer 24

ok, so we can see that for each sales, he earns \(p\), hence \(ps\).

But for each sale (necklace) he has to spend $6, hence\(-6s\)

So overall \(\text{Total Profit} = sp - 6s\).

Make sense so far?

Answer 25

what is p

Answer 26

nvm

Answer 27

what is S

Answer 28

Providing source would be nice @sangya21

I am positive you didn't write it.

Answer 29

s is sales

Answer 30

what did you choose PS?

Answer 31

or more specifically, s is amount of necklace he sold.

Answer 32

why did you choose ps though?

Answer 33

for each necklace he sold, he earn p, the price of necklace, right?

Answer 34

So he would earn sp

Answer 35

yes but profit is revenue-cost correct?

Answer 36

I don't know finance stuffs, lol...

Answer 37

This is calculus lol

Answer 38

But continue

Answer 39

but we basically can see how it work, right?

I mean I don't know exactly what revenue-cost means

Answer 40

you agree with how I set up equation, right?

Answer 41

@micahwood50 apart from revenue cost topic, it is also basic maths saying it is linear thus increase in sales is increase in profit.
Plus I did this in my high school so I remembered.

Answer 42

Do you follow with this equation \(\text{Total Profit} = ps - 6s\)?

Answer 43

yes

Answer 44

One question where did you come up with that problem?

Answer 45

Ok, obvious we have two variables, so we would need to find relationship between s and p, so we can sub in, then equation would only have one variable.

Answer 46

I mean equation

Answer 47

Well, problem said he just sell necklace, earn the price of necklace ( ps ), but for each necklace, he have to spend $6, for materials (-6s).

Nothing else, like tax or whatever

Answer 48

from what problem said, we can see how everything works by ourselves.

Answer 49

okay so how did you get 13

Answer 50

OK, so we need to find relationship between s and p.

here problem said "he sold the necklaces for $10 each and his sales averaged 20 per day. He also found that for each $1 increase in price sales drop by two per day."

So we can see that in equation, there is point \((10, 20)\) (p, s)

and for \((p+1, s-2)\), we also have \((11, 18)\)

then we can develop equation using these points.

Answer 51

hmm, i start to see why there is confusion in that... but somehow I got right answer

Answer 52

okay

Answer 53

I appreciate all your time so here's your medal and now I'm your fan. Ill ask one of my friends

Answer 54

okay, so you get what I tried so far?

or too confused? lol

Answer 55

too confused lmfao

Answer 56

It probably makes sense in your head

Answer 57

I am sure @mathstudent55 can do job better, if you still want helps.

Answer 58

@mathstudent55 if you can that will be great

Answer 59

he left lol.

just start new question and someone else should come.

Answer 60

if you want to