Question

a light is placed on the tip of a windmill blade to protect planes from crashing into it. the windmills center is 400 feet above the ground and the blades of the windmill are 220 feet long. the windmill rotates in the counter - clockwise direction and completes one full revolution every 10 seconds starting from the 3 - oclock position. let t be the time (in seconds) since the windmill started rotating, and let f(t) represent the lights height above the ground (in feet) after t seconds.
a make a drawing that represents the quantities in the problem context and how they are related. upload it as a pdf or jpg. be sure to label everything clearly.
b determine the equation of the midline for function f
c determine the amplitude of function f
d determine the period of function f
e define the formula for function f that represents the lights height above the ground in terms of the amount of time in seconds, t, since the windmill started rotating. graph the function using desmos to check your answer.

• try again make sure you complete all of the previous parts correctly. what is the functions amplitude? the functions midline? the functions period? how do we use this information to write the functions formula?

should we use a cosine function or a sine function? why?

Explanation:

Step1: Determine the midline

The midline is the average of the maximum and minimum values. The maximum value is \(400 + 220=620\) feet and the minimum value is \(400 - 220 = 180\) feet. The midline \(y=\frac{620 + 180}{2}=400\) feet.

Step2: Determine the amplitude

The amplitude is the distance from the midline to the maximum (or minimum) value. \(A=\frac{620-180}{2}=220\) feet.

Step3: Determine the period

The wind - mill completes one full revolution every \(T = 10\) seconds.

Step4: Write the general form of the sinusoidal function

The general form of a sinusoidal function is \(y = A\sin(Bt)+k\) (assuming it starts at the mid - line). We know that \(A = 220\), \(k = 400\), and the formula for the period \(T=\frac{2\pi}{B}\). Since \(T = 10\), then \(B=\frac{2\pi}{10}=\frac{\pi}{5}\).

Answer:

The function is \(y = 220\sin(\frac{\pi}{5}t)+400\)