Require help to solve an Inequality

Question

$|0.022\cdot e^{-2t}\cdot sin(11t)|<0.01$

Values of $t$ for which the inequality holds.

Thanks in advance

Answer 1

Taking into account @marty cohen's answer and using whole numbers, you want to find $t$ such that $$f(t)=22\, e^{-2 t} \sin (11 t)-10<0$$ If you want a more precise result, you could use a numerical method for solving the equation.

The problem is that we need a reasonable guess. We have $$f'(t)=22 e^{-2 t} (11 \cos (11 t)-2 \sin (11 t))$$ It cancels when $$11 \cos (11 t)-2 \sin (11 t)=0 \implies t=\frac 1 {11}\tan ^{-1}\left(\frac{11}{2}\right)$$ but, by the second derivative test, this is a maximum. So, we need to start on the right of it. So , let us use $$t_0=\frac 2 {11}\tan ^{-1}\left(\frac{11}{2}\right)$$ and Newton iterates are

$$\left( \begin{array}{cc} n & t_n \\ 0 & 0.252899 \\ 1 & 0.216374 \\ 2 & 0.215230 \\ 3 & 0.215227 \end{array} \right)$$

The inequality then holds as soon as $t\geq 0.216$.

Answer 2

Since $|\sin(x)| \le 1$, a sufficient condition is $|0.022 e^{-2t}| < 0.01$ or $e^{-2t} < 0.01/0.022=1/2.2$ or $e^{2t} > 2.2$ or $t > \ln(2.2)/2$.