Question

Calculus help
*MEDAL*

Answer 1

I got so far:
V=(16-2x)(12-2x)(x)
V=4x^3-56x^2+192x

Answer 2

x is the height?
ah ok, ya looks good so far :)

Answer 3

You want to figure out what value of x will maximize your volume function.
So now we're interested in `critical values`.

Answer 4

Then do you take V=4x^3-56x^2+192x and get that derivative? or do you set it equal to 0?

Answer 5

Yes, derivative first, figure out the rate of change of the volume function.
And then look for critical values by setting derivative equal to zero.

Answer 6

V'=12x^2-112x+192 correct?

Answer 7

mmm k good

Answer 8

"Your work must include a statement of the function and its derivative"
So would it just be that the volume of the box is 4x^3-56x^2+192x and the derivative is 12x^2-112x+192

Answer 9

Ya, that seems sufficient :)
Looks like teacher just wants to see some of your work.

Answer 10

wait hold on

Answer 11

This one is NOT going to factor nicely, unfortunate :) lol

Answer 12

Yeah i must get the max volume

Answer 13

\[\large\rm 0=12x^2-112x+192\]So figure out your critical points.

Answer 14

Hint: Use Quadratic Formula,
Divide each term by 12 before doing that though!!

Answer 15

oh okay. hold on

Answer 16

4(3x^2-28x+48)

Answer 17

No no no, you can take more than that out ;)
I made the same mistake when I doing it lol.

Take a 12 out of everything.

Answer 18

12 cant be taken out of 112

Answer 19

Oh oh did I make a boo boo? BLahhhh

Answer 20

lol

Answer 21

Ok good so you took a 4 out.
Put the rest of it into the Quadratic Formula.
Let's see if we can come up with the same decimal values.

Answer 22

oh i suck at quadratic formula

Answer 23

I cant get it @zepdrix

Answer 24

you -_-

Answer 25

\[\large\rm ax^2+bx+c\]Leads to solutions through Quadratic Formula,\[\large\rm x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

So we have:\[\large\rm 3x^2-28x+48\]Leading to,\[\large\rm x=\frac{28\pm\sqrt{(-28)^2-4(3)(48)}}{2(3)}\]Understand the setup?

Answer 26

Right, yes i got that far

Answer 27

\[\large\rm x=\frac{28\pm\sqrt{208}}{6}\]So then uhh...
are you having trouble with using your calculator or something? :o

Answer 28

We'll consider both the plus, and minus cases separately.

Answer 29

No i just got that. Lol i solved it.

Answer 30

What solution you get? :)
You should get two solutions.
Do you understand which one is the correct value?

Answer 31

x=2.26296581, 7.07036751

Answer 32

it would be the first one, corrrect?

Answer 33

Ya it looks like x=7.07 doesn't even work with our equation.
Width: (12-2x) = 12-14
Which gives us a negative length for our box! :)

So yesss. Good job.

Answer 34

Which equation would I plug the 2.26 into?

Answer 35

@zepdrix

Answer 36

4x^3-56x^2+192x correct

Answer 37

We used the derivative function to find our maximizer.
Now we want to know the actual maximum volume,
so we're plugging this into the volume function. not the derivative. yes.

Answer 38

so 194.067 cm^3 is the max volume

Answer 39

Ln^3 i mean for cubic inches

Answer 40

Yay good job \c:/