Question

Determine by Lagrange’s formula, the percentage number of criminals under 35 years:

Age % number of criminals

under 25 years 52

under 30 years 67.3

under 40 years 84.1

under 50 years 94.4

Answer 1

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} x & 25 & 30 & 40 & 50\\ \hline y& 52 & 67.3 & 84.1 & 94.4 \\ \hdashline \end{array}$, where $x$ denotes age and $y$ denotes % of criminals.

By Lagrange’s interpolation formula we have:

$f(x)= \frac{(x-x_1)(x-x_2)(x-x_3)}{(x_0-x_1)(x_0-x_2)(x_0-x_3)}y_0+ \frac{(x-x_0)(x-x_2)(x-x_3)}{(x_1-x_0)(x_1-x_2)(x_1-x_3)} y_1+ \frac{(x-x_0)(x-x_1)(x-x_3)}{(x_2-x_0)(x_2-x_1)(x_2-x_3)}y_2+ \frac{(x-x_0)(x-x_1)(x-x_2)}{(x_3-x_0)(x_3-x_1)(x_3-x_2)} y_3$

We put $x=35:$

$y(35)=f(35)= \frac{(35-30)(35-40)(35-50)}{(25-30)(25-40)(25-50)}52+ \frac{(35-25)(35-40)(35-50)}{(30-25)(30-40)(30-50)} 67.3+ \frac{(35-25)(35-30)(35-50)}{(40-25)(40-30)(40-50)}84.1+ \frac{(35-25)(35-30)(35-40)}{(50-25)(50-30)(50-40)} 94.4=-\tfrac{1}{5}\cdot 52+\tfrac{3}{4}\cdot 67.3+\tfrac{1}{2}\cdot 84.1-\tfrac{1}{20}\cdot 94.4=77.405$

Answer: the percentage number of criminals under 35 years is 77.405.