Respuesta :
First of all, it is important to know that the tip of the hour hand and the ceiling varies sinusoidally, which means it has to be represented by a sinusoidal function
[tex]y=A\sin (kx+r)+C[/tex]According to the problem, the amplitude is 15 because that's the length of the hour hand. So, A = 15.
Given that it's about a clock, the period is 12 hours because each lap takes that time. So,
[tex]\frac{2\pi}{k}=12[/tex]We solve for k.
[tex]\begin{gathered} 2\pi=12k \\ k=\frac{2\pi}{12} \\ k=\frac{\pi}{6} \end{gathered}[/tex]We found the constant.
On the other hand, the maximum value takes place when x = 6. So, the maximum value is y = 53. Using the information we have at the moment, we form the following.
[tex]\begin{gathered} 15\sin (\pi+r)+C=53 \\ -15\sin (r)+C=53 \end{gathered}[/tex]Then, we evaluate the function when x = 0, and y = 23 to get another equation and form a system.
[tex]15\sin (r)+C=23[/tex]If we combine the equations, we get
[tex]C=\frac{76}{2}=38[/tex]Therefore, the equation is
[tex]y=-15\cos (\frac{\pi x}{6})+38[/tex]Observe that the function is a cosine because the term r is equal to pi/2.
