I WILL MEDAL PLEASE HELP!!
Photo Necessities produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x.
The company also discovered that it costs $28 to produce 2 camera cases, $50 to produce 4 camera cases, and $140 to produce 10 camera cases.
Find the total cost of producing 12 camera cases.

Question

I WILL MEDAL PLEASE HELP!!
Photo Necessities produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x.
The company also discovered that it costs $28 to produce 2 camera cases, $50 to produce 4 camera cases, and $140 to produce 10 camera cases.  
Find the total cost of producing 12 camera cases.

misty1212 · Answer

HI!!

misty1212 · Answer

you have to find the quadratic function $C(x)=ax^2+bx+c$ that goes through the points $$(2,28),(4,50),(10,140)$$ it is kind of a pain, but we can do it

misty1212 · Answer

how much time you got?

misty1212 · Answer

because we can either solve a 3 by 3 system of equations, or we can cheat

anonymous · Answer

i have all day lol

misty1212 · Answer

lets cheat first since then we can check our answer once we get it

anonymous · Answer

okay ! (:

misty1212 · Answer

http://www.wolframalpha.com/input/?i=quadratic+%282%2C28%29%2C%284%2C50%29%2C%2810%2C140%29

misty1212 · Answer

looks like it is $$C(x)=\frac{1}{2}x^2+8x+10$$ a perfect fit
now want to do it the slow way?

anonymous · Answer

Yes

misty1212 · Answer

you have to evaluate $$C(x)=ax^2+bx+c$$ at the three different value of $x$ 
so we know $$C(2)=4a+2b+c=28$$

misty1212 · Answer

i replaced $x$ by $2$ and set the result equal to 28

misty1212 · Answer

let me know if it is clear how i got that
we got two more to do

anonymous · Answer

its clear (:

misty1212 · Answer

ok now what do you get when you replace $x$ by $4$?

anonymous · Answer

how do i know what to put in the ax + bx spaces ???? when i replace it with 4

misty1212 · Answer

$$ax^2+bx+c=50$$ if $x=4$ so a start is $$a\times 4^2+b\times 4+c=50$$

misty1212 · Answer

lets   back up 
we have $$c(x)=ax^2+bx+c$$ and you are told if $x=2$ then $C=28$ which means $$C(2)=a\times 2^2+b\times 2+c=28$$ or $$4a+2b+c=50$$

misty1212 · Answer

in other words, we replace $x$ by $2$ and set the result equal to 28

misty1212 · Answer

now replace $x$ by $4$ and set the result equal to 50

anonymous · Answer

okay!

misty1212 · Answer

what do you get?

anonymous · Answer

c(4)=a x 4^2 + b x 4 + c = 50

anonymous · Answer

??

misty1212 · Answer

yhes

misty1212 · Answer

which we should write as $$16a+4b+c=50$$

misty1212 · Answer

on account of $4^2=16$ and we usually put the coefficient first, variable second

misty1212 · Answer

finally put $x=10$ and set the result equal to 140

anonymous · Answer

c(10) = a x 10^2 + b x 10 + c = 140

misty1212 · Answer

right, or $$100a+10b+c=140$$

misty1212 · Answer

so now we have to solve the 3 by 3 system of equations $$4a+2b+c=28\16a+4b+c=50\
100a+10b+c=140$$

anonymous · Answer

okay and how do we do that ?

misty1212 · Answer

cry

misty1212 · Answer

it is a real pain
probably using some sort of elimination unless this is a class in matrices

anonymous · Answer

this is algebra 1B lol

misty1212 · Answer

wow your math teacher must hate you!

anonymous · Answer

im in online school ! lol

anonymous · Answer

its harder

misty1212 · Answer

oh in that case it  is a set up for cheating, but lets go ahead and do it

anonymous · Answer

okay!!!

misty1212 · Answer

lets just look at the first two $$16a+4b+c=50\
4a+2b+c=28$$ 
subtract the second line from the first, get $$12a+2b=22$$ since $c-c=0$

anonymous · Answer

okay then what ?

misty1212 · Answer

redo it with the first and last $$100a+10b+c=140\
4a+2b+c=28$$ subtract the second from the first, get $$96a+8b=112$$i hope my arithmetic is right

misty1212 · Answer

now we have taken the 3 by 3 system and turned it in to a 2 by 2 system $$12a+2b=22\
96a+8b=112$$

anonymous · Answer

okay i see !

misty1212 · Answer

so that is not too hard to solve 
maybe multiply the first one by $-4$ all the way across and get $$-48a-8b=-88\
96a+8b=112$$ then add them up

misty1212 · Answer

you get $$48a=22$$ so $a=\frac{1}{2}$ 
now that we have $a$ we can substitute back to find $b$

misty1212 · Answer

i will let you do that yourself, especially since we already know that $b=8$ and $c=10$

misty1212 · Answer

of course you are STILL not done

misty1212 · Answer

once you have $$C(x)=\frac{1}{2}x^2+10x+8$$  you have to find $$C(12)$$ but that is easy

anonymous · Answer

ugh lol okay im gonna try to do it

misty1212 · Answer

try and do which part?

anonymous · Answer

finding c(12)

misty1212 · Answer

$$C(12)$$ is a regular compuation, it is $$\frac{1}{2}(12)^2+10\times 12+8$$

misty1212 · Answer

i can check your answer if you like

anonymous · Answer

what is the answer ? because im confused :( this is making my head hurt lol

misty1212 · Answer

http://www.wolframalpha.com/input/?i=.5%2812%29%5E2%2B10*12%2B8

anonymous · Answer

200??

misty1212 · Answer

yes

anonymous · Answer

thats the complete answer to the whole question???

misty1212 · Answer

yes

anonymous · Answer

Okay thankyou!!!!!!

misty1212 · Answer

$$\color\magenta\heartsuit$$