To determine the angle by which the boat's direction changed when it made its first turn, we first need to recognize that the given distances form a triangle. We can use the law of cosines to solve for the angle.
Given:
- Distance traveled north (a) = 28 miles
- Distance traveled southwest (b) = 25 miles
- Final distance from the starting point to the stopping point (c) = 18 miles
The law of cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
We need to solve for angle A, which represents the change in direction.
1. Substitute the given values into the law of cosines:
[tex]\[ 28^2 = 25^2 + 18^2 - 2 \cdot 25 \cdot 18 \cos(A) \][/tex]
2. Calculate the squares of the distances:
[tex]\[ 784 = 625 + 324 - 2 \cdot 25 \cdot 18 \cos(A) \][/tex]
[tex]\[ 784 = 949 - 900 \cos(A) \][/tex]
3. Rearrange the equation to isolate [tex]\(\cos(A)\)[/tex]:
[tex]\[ 784 - 949 = -900 \cos(A) \][/tex]
[tex]\[ -165 = -900 \cos(A) \][/tex]
[tex]\[ \cos(A) = \frac{165}{900} \][/tex]
[tex]\[ \cos(A) \approx 0.1833 \][/tex]
4. Find the angle A using the inverse cosine function (arccos):
[tex]\[ A = \arccos(0.1833) \][/tex]
5. Convert this value from radians to degrees:
[tex]\[ A \approx 79.44^\circ \][/tex]
6. Round to the nearest degree:
[tex]\[ A \approx 79^\circ \][/tex]
Thus, the direction of the boat changed by approximately 79 degrees when it made its first turn.
Using the context of the problem, the correct answer from the choices provided would be not listed because:
None of the choices {30, 39, 46, 50 degrees} match the calculated approximate value of 79 degrees.
