Answer:
17.7 degree
Explanation:
We can begin by drawing a diagram:
```
C
/|
/ |
/α | 10.5 mi```
A------β------B
\ /
\ /
\/
8.8 mi
```
We can label angles β and α as shown, where β represents the angle from point C to point B and α```
A ------- 9.2 mi -------- B
```
Let's label some angles in the diagram:
- Angle A: the angle between the line from A to C and the line from A to B
- Angle B: the angle between the line from C to A and the line from C to B
- Angle α: the angle at the buoy C
We want to find the measure of angle α. To do that, we can use the law of cosines to find the length of the side opposite angle α:
```
c^2 = a^2 + b^2 - 2ab cos(C)
```
where a = 8.8, b = 10.5, and C = angle B. We can solve for cos(C):
```
cos(C) = (a^2 + b^2 - c^2) / 2ab
```
Since we know a, b, and c (8.8, 10.5, and 9.2 respectively), we can calculate cos(C):
```
cos(C) = (8.8^2 + 10.5^2 - 9.2^2) / (2 * 8.8 * 10.5) ≈ 0.701
```
Now we can find angle B using the inverse cosine function:
```
B = cos^-1(cos(C)) ≈ 45.4°
```
Finally, we can find angle α by subtracting angle A from angle B:
```
α = B - A ≈ 17.7°
```
So the measure of the turn that the ship has to make at the buoy is approximately 17.7 degrees.