Question

Estimate the cube root of the number $40.$

• A. $3.1$
• B. $3.2$
• C. $3.4$
• D. $3.9$

Answer 1

The correct option is C $3.4$

We use the Babylonian Algorithm for cube roots here

According to the algorithm, the cube root is given by the formula

$x_{n+1}=\frac{(2x_n+(N/x_{n^2}\left)\right)}{3}$
where,
• $N$ is the number for which cube root is to be found
• $x_n$ is the initial approximation of the cube root
• $x_{n+1}$ is the subsequent improvement on the cube root
• In this case
$N=40$

$x_0=3$ since $3^3<40<4^3$


$\therefore$ $x_1=\frac{\left(\right(2\times 3)+(40/3^2\left)\right)}{3}=\frac{(6+(40/9\left)\right)}{3}=3.4$


$\Rightarrow x_2=\frac{(2\times 3.4+40/(3.4{)}^2)}{3}=\frac{(6.8+(40/11.56\left)\right)}{3}=\frac{(6.8+3.46)}{3}=3.4$


We can see the value stabilizes around 3.4. Hence, the answer is C.