The blades of a windmill turn on an axis that is 40 feet from the ground. The blades are 15 feet long and complete 3 rotations every minute. Write a sine model, [tex]y = a \sin (b t) + k[/tex], for the height (in feet) of the end of one blade as a function of time [tex]t[/tex] (in seconds). Assume the blade is pointing to the right when [tex]t = 0[/tex] and that the windmill turns counterclockwise at a constant rate.

1. Determine the amplitude, [tex]a[/tex]:
- [tex]a[/tex] is the length of the blades, which is 15 feet.

2. Determine the vertical shift, [tex]k[/tex]:
- [tex]k[/tex] is the height of the axis above the ground, which is 40 feet.

3. Express the values:
- [tex]a = 15[/tex]
- [tex]k = 40[/tex]

Now, complete the sine model with these values:
[tex]y = 15 \sin (b t) + 40[/tex]

To find [tex]b[/tex], note that 3 rotations per minute equate to a frequency of 0.05 rotations per second, resulting in [tex]b = 2\pi \times 0.05 = 0.3\pi[/tex].

We start by noting that the tip of the windmill blade moves in a circular path. The sine model for the height is given by

[tex]$$
y = a \sin(b t) + k.
$$[/tex]

1. Since the blades are 15 feet long, the amplitude (which is the distance from the center of the circle to the tip) is

[tex]$$
a = 15.
$$[/tex]

2. The axis of the windmill is 40 feet from the ground. This means that the center of the circular motion is 40 feet above the ground. Thus, the vertical shift is

[tex]$$
k = 40.
$$[/tex]

3. (For context, although not required for the values asked) The windmill makes 3 rotations per minute, meaning the period is

[tex]$$
T = \frac{60 \text{ seconds}}{3} = 20 \text{ seconds}.
$$[/tex]

Thus, the angular frequency is computed as

[tex]$$
b = \frac{2\pi}{T} = \frac{2\pi}{20} = \frac{\pi}{10}.
$$[/tex]

So, the sine model for the height of the end of one blade is

[tex]$$
y = 15 \sin\left(\frac{\pi}{10}t\right) + 40.
$$[/tex]

In summary:

- The amplitude, [tex]$a$[/tex], is the radius of the circular motion and is 15.
- The vertical shift, [tex]$k$[/tex], is the height of the windmill’s axis, which is 40.

Thus, the answers are:

[tex]$$
a = 15 \quad \text{and} \quad k = 40.
$$[/tex]