A fan with a diameter of 10cm is spinning at 25 revolutions per minute clockwise. The fan's reference blade starts pointing directly upwards. Determine an equation for a sinusoidal function that represents the fan's reference blade's vertical distance from the center (a negative value being accepted). State the period, amplitude and vertical shift for this function.

since t=0 when y is a max, we will have

y = a cos(kt)

the radius is 5, so

y = 5cos(kt)

the period is 1/25 min, so k/2pi = 1/25
k = 2pi/25, and thus

y = 5cos(2pi/25 t)

To determine an equation for a sinusoidal function representing the vertical distance of the fan's reference blade from the center, we'll consider the properties involved: period, amplitude, and vertical shift.

1. Period:
The period of a sinusoidal function refers to the length of one complete cycle. In this case, the fan is spinning at 25 revolutions per minute. To find the period (T) in minutes, we can use the formula:
T = 1 / (number of revolutions per minute)

So, the period (T) for this fan is:
T = 1 / 25

2. Amplitude:
The amplitude (A) represents the maximum absolute value of the vertical distance from the center. In this case, the fan has a diameter of 10 cm, so the radius is 5 cm. The amplitude is half of the diameter, so the amplitude is:
A = 5 cm

3. Vertical Shift:
The vertical shift (C) represents any upward or downward shift of the sinusoidal function. As the reference blade starts pointing directly upwards, there is no vertical shift, so C = 0.

Combining these elements, we can write the equation for the sinusoidal function as:
f(x) = A sin(2π / T * (x + C))

Plugging in the known values:
f(x) = 5 sin(2π / (1/25) * x)

Simplifying further, we get:
f(x) = 5 sin(50πx)

Therefore, the equation for the sinusoidal function representing the fan's reference blade's vertical distance from the center is f(x) = 5 sin(50πx). The period is 1/25 (minutes), the amplitude is 5 cm, and there is no vertical shift (C = 0).