Chapter 8, Problem 18CT

To determine

A function that models a horizontal displacement of a $5$ feet swing after time $t$, where one full swing takes $6$ seconds and at the peak of swing, it made an angle ${42}^0$ with the vertical.

Answer to Problem 18CT

Solution:

$f\left(t\right)=3.3456\cdot \sin \left(\frac{\pi }{3}t\right)$

Explanation of Solution

Given information:

A $5$ feet swing takes $6$ seconds to complete one full swing and at the peak of swing, it made an angle ${42}^0$ with the vertical.

Explanation:

A swing moves in to and fro motion in a periodic time, thus the function of horizontal displacement should be $f\left(t\right)=A\sin \left(Bt\right)$.

A swing takes $6$ seconds to complete one full swing, therefore the period of the function is $6$

$\Rightarrow B=\frac{2\pi }{6}=\frac{\pi }{3}$

Now, find the amplitude of the function by using the following figure.

Here, $A$ is the initial position of the swing. $B$ is the peak position

Construct $BD$ perpendicular to the vertical axis $OA$ and $x$ denotes the horizontal distance between $D$ and $B$

In triangle $ODB$, angle $O=42°$, $D=90°$ and length $OB=5$

By using the sine ratio $\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$,

$\Rightarrow \sin 42°=\frac{x}{5}$

$\Rightarrow x=5\sin 42°=3.3457$

Thus, amplitude of the function is $A=5\sin 42=3.3457$

By substituting values of $A,B$ in $f\left(t\right)=A\sin \left(Bt\right)$

$\Rightarrow f\left(t\right)=3.3457\sin \left(\frac{\pi }{3}t\right)$

Thus, the function that models horizontal displacement of swing is $f\left(t\right)=3.3457\sin \left(\frac{\pi }{3}t\right)$