Question

the cost in cents to produce x cups of Mountain Thunder Lemonade at Junior's Lemonade stand is C(x)=18x+240 x>=0 and the price function in cents per cup is p(x) =90-3x 0<=x<=30
find the price to charge per cup in order to maximize profit and i found 12 cups need to be sold to maximize profit

Answer 1

@Directrix

Answer 2

Does the attachment show the same problem as yours? @ninab731

Answer 3

p(x) =\[-3x^2+72x-240\]
Maximum profit = \[\frac{ -72 }{ -6 }=12\]
\[-3(12)^2+72(12)-240=192 \] $1.92 maximum profit
yes that attachment is correct

Answer 4

Here are the instructions.

Answer 5

@Directrix
http://awesomescreenshot.com/0df1ps169d

Answer 6

I will have to read an example of this type problem and how to solve it. $1.92 for maximum profit seems low.

Answer 7

http://awesomescreenshot.com/0c11ps1fd7

Answer 8

I don't follow how the reasoning behind getting 12 cups for maximum profit. Will you explain that in words?

Answer 9

using the quick vertex formula -b/2a
so -72/-6 gives us 12

Answer 10

then plugged that into the equation to get $1.92

Answer 11

To break even, you have to sell 4 cups or 20 cups.

That comes from finding the zeros of the function f(x) = -3x² + 72x - 240 in the interval [0,30].

Answer 12

Okay, you got the maximum number of cups from the vertex of f(x). Is that correct?

Answer 13

yes thats right

Answer 14

Okay, I get the 12 cups. So, that's 192 cents which is $1.92.

Answer 15

right so how do we find the price to charge per item in order to maximize profit

Answer 16

Where is the cost function?

Answer 17

\[C(x)=18x+240 \]

Answer 18

C(0 cups)=18*0+240 where x>=0

C(0) = $2.40 so it costs $2.40 before you can make any cups of lemonade.

Answer 19

ok so where are they getting it as saying
the price per cup should be set at 54 cents to maximize profit

Answer 20

What is the cost for 12 cups?

Answer 21

oh ok i get it so plug the 12 for C(12)?

Answer 22

I'm just asking as I think. I don't know for sure doing that will give 54 cents. Let me think.

Answer 23

18(12)+240 = 456

Answer 24

Look at this:

the price function in cents per cup is p(x) =90-3x 0<=x<=30

p(12 cups) = 90 - 3*12 = 90 - 36 = ? @ninab731

Answer 25

ah 54!!!

Answer 26

This is a crazy problem and no way would I do all that work to make $1.92 at the lemonade stand.

Answer 27

haha thanks a bunch for the help

Answer 28

You did all the work. It was fun.

Answer 29

be a lot of work to make a 1.92 haha no way lol