How to Calculate the Edge Length of a Cube Given Its Volume and Surface Area?

[zak100]

Homework Statement

The volume of a cube is 'v' cubic yards, and its surface area is 'a' square feet. If 'v' = 'a' what is the length in inches of each edge?

Homework Equations

volume of a cube = e^3
surface area of cube = 6e^2

The Attempt at a Solution

volume of a cube = 'v' cubic yards
= 'v' * 3 feet
= 'v' * 3 * 12 inches
= 36 'v' inches
surface area of a cube = 6 * e^2
'a' = 6 * e ^2
'v' = 6 * e^2

Now volume of cube = 36 * 6 * e ^2 inches
e^3 = 216 e^2 inches
e = 216 inches

But answer is not correct. Some body please guide me.

Zulfi.

 

[berkeman]

volume of a cube = 'v' cubic yards
= 'v' * 3 feet
= 'v' * 3 * 12 inches
= 36 'v' inches
surface area of a cube = 6 * e^2
'a' = 6 * e ^2
'v' = 6 * e^2

It looks like your unit conversions are not correct. Remember that to convert units, you basically multiply by "1"...

1yd^3 = 1yd^3 (\frac{36in}{1yd})^3 = ? in^3

 

[zak100]

Hi,
will it be:

volume of a cube = 'v' cubic yards
= 'v' * (3)^3 feet
= 'v' * 9 feet
...
...

Is the above correct?

Zulfi.

 

[berkeman]

Hi,
will it be:

volume of a cube = 'v' cubic yards
= 'v' * (3)^3 feet
= 'v' * 9 feet
...
...

Is the above correct?

Zulfi.

I'm not sure. The question is a bit strange since they are asking you to equate two numbers, even though they have different units.

The volume of a cube is 'v' cubic yards, and its surface area is 'a' square feet. If 'v' = 'a' what is the length in inches of each edge?

So I would approach this by starting with the variable e for the length of each edge in inches.

Then I would write the equation to give me the volume "v" of the cube in cubic yards based on e, using the appropriate unit conversion from in^3 to yd^3

Then I would write the equation to give me the surface area "a" of a side in cubic feet based on e, using the appropriate unit conversion from in^2 to ft^2

And finally I would set "a" = "v" and solve for e in inches. Can you show how that calculation would go (using the hint that I gave you above for how to handle such unit conversions)?

 

[berkeman]

BTW, I just worked it out that way, and seem to get a consistent answer.

 

[haruspex]

The question is a bit strange since they are asking you to equate two numbers, even though they have different units.

Numbers do not have units. If told the volume is v, that would be a physical quantity so would have dimension and its value would have units. But we are told the volume is v cubic yards. That makes v a dimensionless number.

 

[berkeman]

Yes, agreed. That's the way I interpret it too, it's just a bit hard to deal with for me mentally. But the method that I outlined in post #4 seemed to work fine. I got an answer different than I was expecting, but it seems to check out.

 

[zak100]

Hi,
I got the solution for this problem. It can be done just by using the actual formulas instead of supposing any symbol for the edges. The solution is attached.

Thanks everybody for your time, comments and interest in my problem.

Zulfi.

 

[Dadface]

I got the same answer but differently.
Let the length be x ft which is equal to x/3 yds

Area of faces in sg ft = 6x²
Volume in cubic yards = x³/27
By equating the two x = 6 times 27 = 162ft = 1944 inches

 

[berkeman]

EDIT -- See the post below after this post

Well then I'm confused.

if e = 1944 inches, then

a = 1944^2 in^2 (\frac{1 ft}{12 in})^2 = 26,244 ft^2

v = 1044^3 in^3 (\frac{1 yd}{36 in})^3 = 157,464 yd^3

I got e = 324 inches, which gives a = v = 729. Did I misinterpret the problem?

 

[berkeman]

Area of faces in sg ft = 6x2

This is what I missed. I only considered the area of one of the faces, not the total surface area. Doh!