A cube has a surface area of 6x^2+60x+150. Find the length of the side.

- anonymous

A cube has a surface area of 6x^2+60x+150. Find the length of the side.

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- anonymous

does x denote the length of the side?

- anonymous

ummmmm......I think so?

- anonymous

for a cube of side a, the surface area is 6a^2

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- anonymous

so 6x^2+60x+150 = 6a^2

- anonymous

if the problem just states what you have posted above, we cannot assume that the side of the cube is denoted by x, so thats why i am assuming it to be a.
all you need to do now is get a in terms of x.
so a^2 = x^2+10x+30, or a = sqrt(x^2+10x+30)

- anonymous

so what is the length of a side?

- radar

A cube surface area consist of 6 equal faces.
Step 1. Divide your formula for total surface area by 6 and the new equation will be the surface area of one face. Dividing by 6 Gives you:\[x ^{2}+10x+25\]

- anonymous

whoops as radar says, it is 25, not 30.

- radar

Now factor that and get (x+5) (x=5) \[(x=5)^{2}\]

- anonymous

........

- anonymous

a = sqrt(x+5)^2
or a = x+5

- radar

\[(x+5)^{2}\]n Is what I meant to type

- anonymous

Thanks guys! But, how can I find out the length of a side from that?

- radar

dhatraditya called it the the length of the face is x+5 or the area is (x+5)^2

- anonymous

you can't. You can only find out length of the side in terms of x. So if you know x to be say,5 , then the side is 5+5 or 10.

- anonymous

Sigh......clear as mud. OMG!

- anonymous

OK lets start over.
you are given that a cube has a surface area 6x^2+60x+150. if x is 1, then its surface area is 6(1)^2+60*1+150
so for x = 1, the surface area is 216.
but we also know that the surface area of a cube in terms of the side of the cube is 6*side^2.
so 6 * side^2 = 216 or side^2 = 36 or side = 6.
Do you see the relationship between x and side? side is 6 and x is 1. so side is x+5.
you can try it out with any x and you will always find side to be 5 more than x.
Is that clearer?

- anonymous

Yes. Thanks. But......

- anonymous

in this problem, you just have to show the relationship between the side and x. suppose you had been given a problem which stated the surface area of a cube is 150. find the side of the cube, what would you have done to find the side?
you would have plugged in 6*side^2 = 150 or side^2 = 25 or side =5.
that is exactly what we are doing here. instead of 150, the surface area is 6x^2+60x+150.
so 6*side^2 = 6x^2+60x+150.
simplify it to get side in terms of x.

- anonymous

So........x+?=length of a side?

- anonymous

yes, we are finding out side in terms of x, it could be x+.. or x-.. or x*.. or x/.. depending on what the equation was. here it turned out that side was x+..

- anonymous

Lol, so what is it then?

- anonymous

as we wrote it earlier,
side = x+5

- anonymous

Perfect!

- anonymous

okay let me give you an example. I say that x^2 is the area of a square of side x. what is x?

- anonymous

No clue.

- anonymous

thats because the equation a^2 is use to define a mathematical concept called square of side a. now, in your problem, someone had the brilliant idea to define the surface area of a cube in terms of x as 6x^2+60x+150. But we already have a standard method of defining the surface area of a cube as 6*side^2. so we find out what the x means in terms of the standard that we use to define the surface area of a cube i.e, the side of the cube.

- anonymous

hope that makes sense to you. squares, cubes, etc are just mathematical concepts, and we use certain equations to define these abstract concepts.

- anonymous

I appreciate your insight. Still a bit to wrap my mind around.

- anonymous

don't worry about it. You will understand what I mean as you progress. :)

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