Question

Julia owns a hardware store. For every increase of 15cents in the price of a package of batteries, she estimates that sales decrease by 20 packages per day. The store normally sells 840 packages of batteries per day, at $6.00 per package.
a) Determine an equation for the revenue, y, with x increase of 15cents
b) What is the maximum daily revenue that Julia can expect from the battery sales?
c) How many packages of batteries are sold when the revenue is at a maximum?

Answer 1

for every \(x\) increase the price goes up by \(.15x\) and so the cost is \(6+.15x\) on the other hand for every \(x\) increase sales go down by 20 units from 840 so you sell \(840-20x\) units at a cost of \(6+.15x\) for a total revenue of
\((840-20x)(6+.15x)=-3x^2+6 x+5040\)
vertex (max) is when \(x=-\frac{b}{2a}=-\frac{6}{2\times -3}=1\) so max is at one 15 cent increase, sell for $6.15

Answer 2

Thank you :)

Answer 3

you actually understood it?

Answer 4

no i just made it up, could easily be wrong

Answer 5

i was talking to EvaHeuves

Answer 6

I don't get where your getting −3x2+6x+5040 from ?

Answer 7

Oh never mind..

Answer 8

ok so let's start from the beginning

Answer 9

yess, so we got the whole y=(840-20x)(6+0.15x)

Answer 10

you understood that part? @EvaHeuves

Answer 11

yeah we did it in class together

Answer 12

ok so then you want to foil that out

Answer 13

Yes because didn't you try plugging a number into the equation and then you got your max value

Answer 14

what's wrong with foil? easy mnemonic @satellite73

Answer 15

y=5040+126x-120x-3x^2

Answer 16

when you multiply out you get a parabola with negative leading coefficient, so it opens down and therefore has a maximum at the vertex.
first coordinate of the vertex of any quadratic is always at \(x=-\frac{b}{2a}\) in this case it happened to be 1

Answer 17

ok perfect @JeremySeed , now write that in standard form
and then you can use the formula that @satellite73 used to find the vertex

Answer 18

or y=-3x^2+6x+5040

Answer 19

@EvaHeuves are yo ufollowing?

Answer 20

ok there is nothing wrong with foil.

Answer 21

Yes I am

Answer 22

y=-3(x^2+6x)+5040
y=-3(x^2+6x+9-9)+5040
y=-3(x^2+6x+9)+5,067
y=-3(x+3)^2+5067

Answer 23

good job @JeremySeed
so now in this equation, what do x and y represent?

Answer 24

somethings wrong cause that means the vertex is (-3,5067) meaning that when x is -3 the max value is reached

Answer 25

you didn't factor the -3 out of the 6x

Answer 26

ahhhhhh I get fully understand now

Answer 27

how do you factor the -3 out of the 6x?

Answer 28

it would be y=-3(x^2+2x)-5040

Answer 29

oh daaaaaaaaaaaaaaaaaaaaaaaamnnnnnnnnnnnnnnnnnnnnnnnnnnnit

Answer 30

k well we got the right idea on how to do it thats all that matters :)

Answer 31

Yes, good job :)

Answer 32

good job to you guys :)
y'all did the math

Answer 33

Thank you, and yes FOILING does help

Answer 34

wow i didn't know this was so hard.
forget this complete the square nonsense when you want the vertex (although it is very handy when you want to solve a quadratic)
use the simple \(-\frac{b}{2a}\) as the \(x\) coordinate of the vertex, and for the \(y\) coordinate, substitute \(-\frac{b}{2a}\) for \(x\)

Answer 35

we are in grade 10 sorry bud havent learned how to do that yet

Answer 36

we have to do it the long way :/

Answer 37

?? do it with \(ax^2+bx+c\) and see that you always get the same thing, namely \(-\frac{b}{2a}\)

Answer 38

Yeah and you lost me.

Answer 39

@satellite73 they have a test tomorrow, and it's best they do what their teacher taught them - not entirely sure if they get credit for work or not but either way go with what you already know guys for now

Answer 40

@cerezas thank you so much for your help, much appreciated.

Answer 41

then if you want to impress your teacher (the one that foisted "foil" on you) you can always write the answer as \(y=a(x-\frac{b}{2a})^2+\text{whatever}\) and you get the "whatever" by substitution

Answer 42

id rather not try to impress my teacher, but forget how to do it and screw up, on something worth 10% of my final mark

Answer 43

@EvaHeuves anytime, honestly
and @satellite73 i agree it's a useful equation, now's just not the time

Answer 44

@cerezas do you realize you have been helping me for the past 4 hours? You have no idea how much I appreciate all your help and how thankful I am for people like you :) you've actually been the biggest help i could ask for so thank you so much <3 :D

Answer 45

couldnt*

Answer 46

wait no could* :p

Answer 47

@JeremySeed don't worry about it, i love doing this. i'm here whenever you need me seriously, i love doing this
tell me how it goes tomorrow!

Answer 48

same goes for you @EvaHeuves

Answer 49

it's always great to help people who want to understand how to work through a problem and not necessarily just to get the answer <3

Answer 50

yeah ill be sure to let you know! im sure it will be a lot better than it would have been if you never helped me :D

Answer 51

awesome! like i said, i'm here all the time, and i can help in other subjects if you need it :D

Answer 52

:) :) Did you like come from Heaven ? Like omg, I thought I was going to die but now I fully understand all the questions that you and @JeremySeed answered together. Thanks again :) <3

Answer 53

glad to be of service! :)

Answer 54

waaaait do you think youre up for answering one more? it was one i already asked and when the people answered i wasnt really clear on it..

Answer 55

yeah sure just post it :)