Solve Cube Volume Conversion Problem: Get Right Answer!

[Ester]

A cube with a volume of one cubic centimeter (1 cc) has equal sides of 1 centimeter squared with a total surface area of 6 square centimeters (= 6 cm2 or 600 mm2). Dividing the same volume into 1000 cubic millimeter units would lead to a total surface area of ?
A. 6 cm2
B. 6 mm2
C. 60 cm2
D. 60 mm2
E. 600 cm2

The right answer is 60 cm2. I don't know how to derive this answer. When I work it out I get 6cm2. This is because
one cubic centimeter (1 cc) / 1000 cubic millimeter units =
1cm3/1000cm3 = 1
and to get the surface area, multiply by 6.

What am I doing wrong, anyone know how to get the right answer?

 

[Tide]

You are dividing the cube into 1,000 little cubes each 0.1 cm across. Therefore each has surface area equal to 6 times 1/100 cm^2 and multiplying by 1,000 gives 60 cm^2.

 

[Ester]

I still don't get it, how you're getting those numbers. I get 0.01cm acroos.

 

[Tide]

10 X 10 X 10 = 1000: you create 1,000 pieces by cutting each linear dimension into 10 parts.

Thus, each little cube is 1/10 cm X 1/10 cm X 1/10 cm (1 mm X 1 mm X 1 mm)

 

[Ester]

Thanks for your help.
However, I'm still a bit unclear on another part.
ok, I've got everything you've said except the part where you multiply by 1000 in the end. Why did you do that. I mean, (.01cm2)(6) should be the surface area, it has the units correct. How do you know when to multiply by 1000, what if you multiple by 100? I just want to know how you get that number.

 

[Ester]

Can somebody help me?

 

[Tide]

Ester,

Obviously, the total volume is unchanged. You are told that the original cube is divided into 1,000 pieces and, clearly, each of those pieces will have a volume of 1 cubic millimeter. The total surface area of the 1,000 little cubes is greater than the original cube because the act of cutting the cube exposes surface that was not there prior to cutting.

Here's another approach you can take. If x is the length of a side of a cube then its volume is $V = x^3$ and its surface area is $S = 6 x^2$. If the volume of a little cube is 1/1000 cc then $\frac {1}{1000} cc = x^3$ so $x = \frac {1}{10} cm$. Therefore, the surface area of each little cube is $S = 6 \times \frac {1}{100} cm^2$. Finally, multiply by 1,000 to get the total surface area of ALL the little cubes!

 

[Ester]

Finally, multiply by 1,000 to get the total surface area of ALL the little cubes![/QUOTE]

That is what I was confused about in addition to the volume being constant. Thanks a lot. :) I really get it now. The second approach helped a lot.

 

[hotvette]

Personally, I think the wording of the problem is really goofy. It would have been much better to say that the same volume was broken in 1000 cubes. Adding the 'cubic mm units' really throws things off. Also, mentally dividing a cube into smaller cubes changes the surface area naught unless the cubes are phsically separated, which it seems was indeed the question. Wasn't clear at all in my opinion.