Question

If the volume of the cube is \[1728c{{m}^{3}},\] the length of its edge is equal to
(a) 12 cm
(b) 14 cm
(c) 16 cm
(d) 24 cm

Answer 1

Hint: To solve the given question, we will first find out what a cube is and what the formula to calculate the volume of the cube is. After doing this, we will assume that the side length of a cube is ‘s’ cm. Then we will apply the formula for the volume of the cube. On doing this, we will get the equation, \[s=\sqrt[3]{1728}.\] To find the cube root of 1728, we will use the prime factorization method. In this method, we will write 1728 as the product of prime numbers and then put this product in the cube root to get the value of s.

Complete step-by-step answer:
Before we start to solve the given question, we must know what a cube is. A cube is a three – dimensional object bounded by six square faces or sides with three meetings at each vertex. Now, we will assume that the length of each side of the cube is ‘s’ cm.

Now, we know that the volume of the cube is given by the formula shown below.
\[\text{Volume}={{\left( \text{side} \right)}^{3}}\]
In our case, \[\text{Volume}=\text{1728 c}{{\text{m}}^{3}}\] and side is s cm. Thus, we will get,
\[1728c{{m}^{3}}={{s}^{3}}\]
\[\Rightarrow {{s}^{3}}=1728c{{m}^{3}}\]
On taking the cube root on both the sides of the above equation, we will get the following equation
\[\sqrt[3]{{{s}^{3}}}=\sqrt[3]{1728}cm\]
\[\Rightarrow s=\sqrt[3]{1728}cm.......\left( i \right)\]
Now, we will find the cube root of 1728 using the prime factorization method. In this method, we will try to write 1728 as the product of the prime numbers. Thus, we have,

Thus, we can say that,
\[1728=2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\]
\[1728={{2}^{6}}\times {{3}^{3}}......\left( ii \right)\]
Now, we will put the value of 1728 from (ii) to (i). Thus, we will get,
\[\Rightarrow s=\sqrt[3]{{{2}^{6}}\times {{3}^{3}}}cm\]
\[\Rightarrow s={{2}^{2}}\times 3\text{ }cm\]
\[\Rightarrow s=4\times 3\text{ }cm\]
\[\Rightarrow s=12\text{ }cm\]
Hence, option (a) is the right answer.

Note: The prime factorization method with the help of which we have found out the cube root of 1728 is not valid every time. We can use this method only when the number of which we have to find the cube root is the perfect cube of an integer. Do not make the silly mistake of finding the square root instead of the cube root of the volume and reporting it as the side of the cube.