Help with Word Problem

A

allie193

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Olivia designs and sells necklaces. The material for each necklace costs $4. She sells an average of 40 necklaces a week for $8 each. She would like to raise the price, but for each $0.50 increase she would lose 4 sales each week.
a) Write an expression to represent the price of each necklace
b) Write an expression to represent the number of necklaces sold per week.
c) Use your answers in a) and b) to write an expression that represents Olivia's revenue from the sales of her necklaces.
d) Write an expression to represent the cost of the materials for the necklaces.
e) Use your answers from c) and d) to write an equation for Olivia's profit
f) What selling price will give Olivia the largest profit?
g) What will her profit be?
h) How many necklaces must she sell to earn this profit?

There was a similar example that covered a-c of this question so I tried to do it the same way to start and then I'm not sure what to do or if that is the right way to do a-c, I used x as the number of price increases:

a) Price=8+0.5x
b) Sales=40-4x
c) Revenue=(8=0.5x)(40-4x)
d) Cost=4(40-4x) ???
e) Profit=Revenue-Cost ???

And I'm not sure if these are correct and if they are what to do next? :oops: Please help! Thank you! :)
 
allie193 said:
Olivia designs and sells necklaces. The material for each necklace costs $4. She sells an average of 40 necklaces a week for $8 each. She would like to raise the price, but for each $0.50 increase she would lose 4 sales each week.
a) Write an expression to represent the price of each necklace
b) Write an expression to represent the number of necklaces sold per week.
c) Use your answers in a) and b) to write an expression that represents Olivia's revenue from the sales of her necklaces.
d) Write an expression to represent the cost of the materials for the necklaces.
e) Use your answers from c) and d) to write an equation for Olivia's profit
f) What selling price will give Olivia the largest profit?
g) What will her profit be?
h) How many necklaces must she sell to earn this profit?

There was a similar example that covered a-c of this question so I tried to do it the same way to start and then I'm not sure what to do or if that is the right way to do a-c, I used x as the number of price increases:

a) Price=8+0.5x<<< Correct
b) Sales=40-4x<<< Correct
c) Revenue=(8+0.5x)(40-4x)<<< Correct
d) Cost=4(40-4x) <<< Correct
e) Profit=Revenue-Cost = P(x) = (8+0.5x)(40-4x) - 4(40-4x)

P(x) is a quadratic equation. If you plot it - you'll get a parabola. Locate the vertex of the parabola to find 'x' for maximum profit.

And I'm not sure if these are correct and if they are what to do next? :oops: Please help! Thank you! :)
Click to expand...
 
Thanks for the reply, that is what I had for the profit formula but I typed the words instead b/c it seemed wrong since I don't know how to plot it. Can you explain that part more? I am likely overthinking it since I tend to do that so I'm sorry if it is really simple and I should know! :oops:
 
allie193 said:
Thanks for the reply, that is what I had for the profit formula but I typed the words instead b/c it seemed wrong since I don't know how to plot it. Can you explain that part more? I am likely overthinking it since I tend to do that so I'm sorry if it is really simple and I should know! :oops:
Click to expand...

Have learned about equation of parabola and its vertex?
 
Yes I have but I haven't had any in a format like that, I'm doing a correspondence math course so I'm trying to figure most of it out on my own using examples and none of them have been in that format so I'm not sure how to figure out the vertex in that format, do I work it out or is it possible to figure the format out in it's current format is I guess what I mean.
 
First multiply it out and get in the format:

\(\displaystyle P(x) \, = \, A\cdot x^2 \, + \, B\cdot x \, + \, C\)

That would be standard format for quadratic equation.

From there - perform "completing the square" operation and get in the following form:

\(\displaystyle P(x) \, = \, A\cdot (x \, - h)^2 \, + \, k\)

then (h,k) is your vertex - and h is the value of 'x' where P(x) is maximum (=k).
 
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