Given that
The period is 24 seconds
The average height is 8 feet.
The distance between the maximum and minimum height is 4 feet.
Cosine function for this model.
At t=0 the function is in average height.
Let y be the height.
Let t be the time of seconds
The general cosine equation is
[tex]y=a\cos (bt)+c[/tex]Here amplitude |a| represents the half distance between the maximum and minimum height.
[tex]|a|=\frac{4}{2}=2[/tex][tex]a=\pm2[/tex]Substitute a=2 in the general equation, we get
[tex]y=2\cos (bt)+c[/tex][tex]\text{Period =}\frac{2\pi}{|b|}[/tex]Substitute period =24, we get
[tex]\text{24 =}\frac{2\pi}{|b|}[/tex]Using the cross product, we get
[tex]24|b|=2\pi[/tex]Dividing both sides by 24, we get
[tex]|b|=\frac{2\pi}{24}=\frac{\pi}{12}[/tex][tex]b=\pm\frac{\pi}{12}[/tex]Substitute b=pi/12 in the general equation, we get
[tex]y=2\cos (\frac{\pi t}{12})+c[/tex]When t=0 the height is 8,
[tex]8=2\cos (\frac{\pi(0)}{12})+c[/tex][tex]8=2(1)+c[/tex][tex]c=8-2=6[/tex][tex]c=6[/tex]Substitute c=6 in the equation, we get
[tex]y=2\cos (\frac{\pi t}{12})+6[/tex]We get the equation for the average height, it will increase upward and reach the highest height.