Question

Alright. There's this cube where the base is represented by the length of x and the width of x, while it's height is represented by the variable h. Now picture that one of the base faces (x * x) is taken away to leave an open-topped box.
This open-topped box has a surface area of 108 cm^2. I've been asked to solve for the variable h, representative of the height.
I created the formula 108=x^2+4xh to solve for this, but I don't know how to solve this equation.

Answer 1

I at first tried substitution by moving x^2 to the other side and then multiplying all pieces by -1/4h, but something didn't seem quite right so I halted that.

Answer 2

Can you rewrite the equation for me real quick just so we can both see it on it's own?

Answer 3

108=x^2+4xh

Answer 4

That gives me one x times x for the single base and 4 x times h for the 4 sides

Answer 5

OK interesting... Are we trying to solve for the variable for the equation?

Answer 6

Yes, solving for h.
Also, I'm being called out of school so I may not respond for a hot second.

Answer 7

I think I did something terribly wrong, but my math teacher never really looks at it fully and kind of glimpses over it despite me needing help.

Answer 8

Okay you are doing a great job so far. What we would do is flip the equation so can you show me what that would look like?

Answer 9

Flip the equation? as is multiply everything by -1?

Answer 10

4hx+x2=108 Like this.<<

Answer 11

Ah, I see. So why are we flipping it if the commutative property allows for you to do it either way?

Answer 12

Do you think it would be more simple to do it the other way. I just prefer this way because it makes more sense to me

Answer 13

Ye, I like having it the other way.

Answer 14

Oh ok, you are going to have to explain how to do it the other way. I am more experienced with the way I am doing it.

Answer 15

Well, it's more the way I say it that makes sense to me, saying that y is then everything that follows is y

Answer 16

anyways

Answer 17

I get 1/4x^3 + 27x = h

Answer 18

I feel as if I got this result incorrectly.

Answer 19

Ok so the way I do it when you flip it, I got h=−x^2+108/4x

Answer 20

wait, did you divide -x^2 by 4x?

Answer 21

Because to my knowledge you divide the entire other side, which is why I just multiplied by a reciprocal.

Answer 22

Let me type it out real quick

Answer 23

So this is how I did it...

4hx+x2=108

4hx+x2+−x2=108+−x2
4hx=−x2+108/4hx
4x=−x2+108/4x
h=−x2+108/4x

Answer 24

I don't know if that is completely right

Answer 25

In the second to third step, where did h go?

Answer 26

Let me right the 3rd part again

Answer 27

4hx=−x2+108

Then this: 4hx/4x=−x2+108/4x

Answer 28

I think that makes more sense

Answer 29

Yeah, but -x^2 also has to get divided by 4x, too, no?

Answer 30

Gah, I'm sorry if I can't understand.

Answer 31

No no, don't apologize for not understanding. Math isn't my strong suite here either.

Answer 32

Here, let me show you how I did it if it helps any

Answer 33

Yes that would help

Answer 34

Given: 108 = x^2 + 4xh
subtraction: -x^2 + 108 = 4xh
Multiplication by reciprocal of -1/4x: 1/4x^3 -27x = h

Answer 35

-h*

Answer 36

so then -1/4x^3 + 27x = h so we don't have a -h

Answer 37

I am not an expert on that way that you did it there so can you explain to me how that would make sense.

Answer 38

Are you supposed to subtract from both sides or just one?

Answer 39

both sides; sorry I didn't touch up on that detail

Answer 40

@hero can you help this user out with his math question because I am a little stuck on how to do the math.

Answer 41

@jaynater could you provide a screen shot of your question so hero could help better understand your question so he could help you.

Answer 42

I'm at school as of now, so that's why I didn't put it here and instead explain the problem to the best of my ability.

Answer 43

I'll be out in another hour give or take, I'll get an upload then.

Answer 44

Yes screenshot is best

Answer 45

The key to this question is that we're dealing with a CUBE


You have a cube that's an open box so it has five sides. Each face of the cube is going to have the same area which is x^2.

The surface area is going to be the sum of the area of those five sides.

Can you now write the equation to solve for the length of the side (which is going to be equal to the hight of the cube)?

Answer 46

Got a screenshot.

Answer 47

You also have help now

Answer 48

But that is why I ask for screenshots. What you posted originally wasn't even the complete question

Answer 49

Ah, so it's not really a cube.

This is an optimization question.

You have so far

108 = x^2 + 4hx

Can you solve for h?

(Just subtract x^2 and then divide by 4x to isolate 'h')

What do you get?

Answer 50

-1/4x+27x=h

Answer 51

I just didn't think h would equal this.

Answer 52

I think you meant to write

-1/4x + 27/x = h

or if we just rearrange it, we would get

\(h = \dfrac{27}{x} - \dfrac{x}{4}\)

Answer 53

That would answer part B

For part C, how would you find the volume of this box?

First, just write the formula for volume using x and h

volume = length * width * height

and you know the base has sides of x (the length and width are the same since it's a square)
and the height is h

so volume = ???

Answer 54

and then we want to take the equation we got from part B where

\(h = \dfrac{27}{x} - \dfrac{x}{4}\)

and we want to replace the h in the volume equation with \(\dfrac{27}{x}-\dfrac{x}{4}\)

can you do that?