Question

The volume of a cubical box is 32.768 cubic metres. The length of one side of the cube is:
[a] 3.2m
[b] 4.2m
[c] 3.0m
[d] 2.8m

Answer 1

Hint: Use the property that if “a” is the length of a side of a cube, then the volume of the cube is ${{a}^{3}}$. Equate this with the volume of the cube. Solve the cubic by taking a cube-root on both sides. Find the cube root of the terms on the RHS of the equation.

Complete step-by-step answer:
Let the side of the cube be x metres.
So, we have the volume of the cube $={{x}^{3}}$cubic m.
Given that volume of cube = 32.768 cubic m.
Hence we have
${{x}^{3}}=32.768$
Taking cube-root on both sides, we get
$x=\sqrt[3]{32.768}$
Finding the cube root of 32.768:
Convert 32.678 in $\dfrac{p}{q}$ form, where p and q are integers and q is non-zero. Note that this can be done since 32.678 is a rational number.
$32.768=\dfrac{32768}{1000}$
Reduce the fraction in lowest terms.
For reducing the fraction in lowest terms, we need to find the HCF of 32678 and 1000
We have
$\begin{align}
 & 32678=1000\times 32+768 \\
 & 1000=768+232 \\
 & 768=232\times 3+72 \\
 & 232=72\times 3+16 \\
 & 72=16\times 4+8 \\
 & 16=8\times 2+0 \\
\end{align}$
Hence gcd(32678,1000) = 8.
Dividing numerator and denominator by 8, we get
$\dfrac{32768}{1000}=\dfrac{4096}{125}$
Writing prime factorisation of 4096
$4096={{2}^{12}}$
Writing prime factorisation of 125
$125={{5}^{3}}$
Hence we have
$\sqrt[3]{32.768}=\sqrt[3]{\dfrac{4096}{125}}=\sqrt[3]{\dfrac{{{2}^{12}}}{{{5}^{3}}}}=\dfrac{{{2}^{4}}}{5}=3.2$
Hence the side of the cube is 3.2 m
Hence option [a] is correct.

Note: Verification.
The volume of the cube with side 3.2 m $=3.2\times 3.2\times 3.2=32.678$cubic m.
Hence our answer is correct.
[2] In the question we have used $\sqrt[n]{\dfrac{p}{q}}=\dfrac{\sqrt[n]{p}}{\sqrt[n]{q}}$