Question

A cube whose volume is $\dfrac{1}{8}$ cubic centimetre is placed on top of a cube whose volume is $1$ cm3. The two cubes are then placed on top of a third cube whose volume is $8$ cm3. The height of the stacked cubes is
A) $3.5$cm
B) $3$cm
C) $7$cm
D) None of these

Answer 1

Hint: We are given the volume of the three cubes. So, by using the formula for volume of a cube we can easily find the length of the side of the cube. Then it is given to us in the question, that they are then stacked in top of each other, and we are to find the total height of the stack. So, it is obvious that the total height will be equal to the sum of the heights of the three cubes, which are equal to the length of the sides of the cubes.

Complete step by step answer:
Given the volumes of the cubes are $\dfrac{1}{8}$cm3, $1$cm3, and $8$cm3.
Now, we know, the formula for volume of a cube is, $V = {a^3}$.
Now, let the sides of the first, second and third cubes be ${a_1},{a_2},{a_3}$ respectively.
Therefore, we can write, for the first cube,
${V_1} = a_1^3$
$ \Rightarrow a_1^3 = \dfrac{1}{8}$
Taking cube root on both sides, we get,
$ \Rightarrow {a_1} = \dfrac{1}{2}$cm
Therefore, we can write, for the second cube,
${V_2} = a_2^3$
$ \Rightarrow a_2^3 = 1$
Taking cube root on both sides, we get,
$ \Rightarrow {a_2} = 1$cm
Therefore, we can write, for the third cube,
${V_3} = a_3^3$
$ \Rightarrow a_3^3 = 8$
Taking cube root on both sides, we get,
$ \Rightarrow {a_3} = 2$cm
[Since, we are talking about length here, we will only take the positive values of the cube roots]
Therefore, the length of sides of the cubes are $\dfrac{1}{2}$cm, $1$cm, and $2$cm respectively.
Now, we are asked to find the total height of the stack after the cubes are placed on top of each other.
Clearly, the total height of the stack will be equal to the sum of the heights of the cubes.
We know, in a cube the height of the cube is equal to the length of side of the cube.
Therefore, the heights of the cubes are $\dfrac{1}{2}$cm, $1$cm, and $2$cm respectively.
Now, total height of the stack is, $H = {h_1} + {h_2} + {h _3}$
Now, putting the values, we get,
$ \Rightarrow H = \dfrac{1}{2} + 1 + 2$
$ \Rightarrow H = 0.5 + 3 = 3.5$cm
Therefore, the total height of the stack is $3.5$cm.

Thus, the correct option is A.

Note:
Here, the most prominent mistake we can do is that, we may add the volumes of the three cubes, than find the height of the cube. But this will be the case that we have melted the three cubes and made another cube, but our case was different.