After t minutes, the height h meters above the ground of the seat is given by h=74+acosbt. How do you show that the period of h is 25 minutes?

Jul 21, 2018

In order for the period to be $= 25 m n$, the value of $b = 0.25$

Explanation:

The function is

$h \left(t\right) = 74 + a \cos b t$..........................$\left(1\right)$

For a periodic function of period $T$

$h \left(t\right) = h \left(t + T\right)$

$h \left(t + T\right) = 74 + a \cos b \left(t + T\right)$

$= 74 + a \cos \left(b t + b T\right)$

$= 74 + a \cos b t \cos b T - a \sin b t \sin b T$.............$\left(2\right)$

Comparing this equation to equation $\left(1\right)$

$\left\{\begin{matrix}\cos b T = 1 \\ \sin b T = 0\end{matrix}\right.$

$\iff$, $b T = 2 \pi$

$\iff$, $T = \frac{2}{b} \pi$

In order for $T = 25 m n$

$\frac{2}{b} \pi = 25$

$\implies$, $b = \frac{2 \pi}{25} = 0.25$