Question

note: figure not drawn to scale.
the figure shows a robotic arm rotating in a circular counterclockwise direction that completes one rotation every 2 seconds. point s is on the tip of the arm, and point x does not move. as the arm rotates at a constant speed, the height of s above x periodically increases and decreases. at time t = 0 seconds, s is at its lowest position, 6 inches directly below x. at its highest position, s is 20 inches directly above x.
the sinusoidal function h models the height of s above x, in inches, as a function of time t, in seconds. a positive value of h(t) indicates s is above x; a negative value of h(t) indicates s is below x.
h: height of s above x

Explanation:

Step1: Determine the amplitude

The amplitude \( A \) is half the distance between the maximum and minimum values. The maximum height is \( 20 \) inches above \( X \), and the minimum height is \( -6 \) inches (6 inches below \( X \)). So, the distance between them is \( 20 - (-6) = 26 \) inches. Then, the amplitude \( A=\frac{26}{2}=13 \) inches.

Step2: Determine the vertical shift (midline)

The midline \( k \) is the average of the maximum and minimum values. So, \( k=\frac{20 + (-6)}{2}=\frac{14}{2} = 7 \) inches.

Step3: Determine the period and angular frequency

The period \( T \) is 2 seconds (time for one rotation). The formula for the angular frequency \( \omega \) is \( \omega=\frac{2\pi}{T} \). Substituting \( T = 2 \), we get \( \omega=\frac{2\pi}{2}=\pi \).

Step4: Determine the phase shift

Since the function starts at its minimum (lowest position) at \( t = 0 \), this is a cosine function reflected over the x - axis (because the standard cosine function starts at its maximum). The general form of a sinusoidal function is \( h(t)=A\cos(\omega(t - \phi))+k \) or \( h(t)=A\sin(\omega(t - \phi))+k \). For a cosine function starting at minimum, we can use \( h(t)=-A\cos(\omega t)+k \) (since reflecting \( \cos(\omega t) \) over the x - axis gives a function that starts at its minimum).

Substituting \( A = 13 \), \( \omega=\pi \), and \( k = 7 \), we get the function \( h(t)=- 13\cos(\pi t)+7 \). Let's verify:

• At \( t = 0 \): \( h(0)=-13\cos(0)+7=-13\times1 + 7=-6 \), which matches the minimum height at \( t = 0 \).
• At \( t = 1 \) (half - period, since \( T = 2 \)): \( h(1)=-13\cos(\pi)+7=-13\times(-1)+7 = 13 + 7=20 \), which matches the maximum height.

Answer:

The sinusoidal function is \( h(t)=-13\cos(\pi t)+7 \) (or an equivalent sine function form, but this cosine - based form is derived from the given conditions). If we were to find specific values (e.g., height at a certain time) or analyze the function further, we can use this formula. For example, if we want to find the height at \( t = 0.5 \) seconds:
\( h(0.5)=-13\cos(0.5\pi)+7=-13\times0 + 7 = 7 \) inches (which is the midline, as expected at the mid - point of the minimum and maximum in a period of 2 seconds, at \( t = 0.5 \) which is a quarter - period from \( t = 0 \) for a period of 2 seconds, and the motion is symmetric).