a. To find the number of degrees corresponding to the range of angle θ, we can use the fact that 1 radian is equal to 180 degrees. So, the range of angles in radians can be converted to degrees by multiplying by 180/π. In this case, the range is 0.4 radians (0.2 + 0.2), so the equivalent range in degrees is (0.4 * 180/π) degrees.
b. To find the arc length on the cathead corresponding to the range of angles, we can use the formula for finding the length of a circular arc. The formula is: arc length = radius * angle (in radians). In this case, the radius is given as 8 ft and the range of angles is 0.4 radians. So, the arc length on the cathead is 8 ft * 0.4 radians.
c. To find the amplitude of the sinusoidal variation in distance d, we can use the fact that the amplitude of a sinusoid is half the range of variation. In this case, the range of angles is given as 0.2 radians, so the amplitude of the sinusoid is 0.2/2 radians.
d. Based on the given information, we can sketch the graph of this sinusoid. First, we know that point P is at its highest point above the ground when t = 0 s. Since the walking beam is horizontal at this point, P is 7 ft above the ground. We also know that P is at its next low point 2.5 s after that. From this information, we can conclude that the period of the sinusoid is 2.5 s. We can now sketch the graph of the sinusoid using the amplitude and period.
e. To find a particular equation expressing d as a function of t, we need more information about the specific form of the sinusoidal variation. Without that information, we cannot provide a particular equation.
f. To find how far above the ground P is at t = 9, we need the equation expressing d as a function of t. Without that equation, we cannot determine the value of d at t = 9.