Question

If a cube of material is subdivided into 1000 smaller cubes of equal size, the surface area of the smaller cubes will be what fraction of the original? How many times greater than the original will the total surface area of the cube be?

Answer 1

Answer on Question #58463 – Chemistry – Other

Task:

If a cube of material is subdivided into 1000 smaller cubes of equal size, the surface area of the smaller cubes will be what fraction of the original? How many times greater than the original will the total surface area of the cube be?

Solution:

large cube

small cube

Let the side of a large cube (starting material) is equal to $\mathbf{a}$ , then the original area of the surface of the cube:

$$S _ {s} (l \arg e c u b e) = 6 \times a ^ {2}.$$

Then how the volume of the large cube will be as follows:

$$V _ {l} = a ^ {3}.$$

Since a large cube is crushed by 1000 small cubes of equal size, the volume of each of the cube will be:

$$V _ {s} = \frac {a}{1 0 0 0} ^ {3}.$$

Therefore, the side of a small cube will be equal to:

$$\sqrt[3]{\frac{a}{1000}}^{3} = \frac{a}{10}.$$

Then the surface area of a small cube:

$$S_{s} \text{ (small cube)} = 6 \times \left(\frac{a}{10}\right)^{2} = \frac{6 \times a^{2}}{100}.$$

Total of such (small) blocks 1000. So, their summary the surface area will be:

$$\sum S_{s} \text{ (small cube)} = 1000 \times S_{s} \text{ (small cube)} = 1000 \times \frac{6 \times a^{2}}{100} = 60 \times a^{2}.$$ $$\frac{S_{s} \text{ (small cube)}}{S_{s} \text{ (large cube)}} = \frac{6 \times a^{2}}{100} / 6 \times a^{2} = \frac{1}{100}.$$ $$\frac{\sum S_{s} \text{ (small cube)}}{S_{s} \text{ (large cube)}} = \frac{60 \times a^{2}}{6 \times a^{2}} = 10.$$

**Answer:**

Thus, the surface area of one small cube will be 1/100 of the surface area of the large cube, and the surface area of all the small cubes 10 times more the surface area of the large cube.

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