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Question 17 of 50
2 Points

Three ships, [tex]A[/tex], [tex]B[/tex], and [tex]C[/tex], are anchored in the Atlantic Ocean. The distance from [tex]A[/tex] to [tex]B[/tex] is 36.318 miles, from [tex]B[/tex] to [tex]C[/tex] is 37.674 miles, and from [tex]C[/tex] to [tex]A[/tex] is 11.164 miles. Find the angle measurements of the triangle formed by the three ships.

A. [tex]m \angle A = 88.28267^\circ, \, m \angle B = 17.22942^\circ, \, m \angle C = 74.4879^\circ[/tex]
B. [tex]m \angle A = 17.22942^\circ, \, m \angle B = 74.4879^\circ, \, m \angle C = 88.28267^\circ[/tex]
C. [tex]m \angle A = 88.28267^\circ, \, m \angle B = 74.4879^\circ, \, m \angle C = 17.22942^\circ[/tex]
D. [tex]m \angle A = 74.4879^\circ, \, m \angle B = 17.22942^\circ, \, m \angle C = 88.28267^\circ[/tex]



Answer :

GinnyAnswer
To determine the angles of the triangle formed by ships [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], we can use the Law of Cosines. The distances between the ships form a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:

- [tex]\(a = 37.674\)[/tex] miles (distance between [tex]\(B\)[/tex] and [tex]\(C\)[/tex])
- [tex]\(b = 11.164\)[/tex] miles (distance between [tex]\(C\)[/tex] and [tex]\(A\)[/tex])
- [tex]\(c = 36.318\)[/tex] miles (distance between [tex]\(A\)[/tex] and [tex]\(B\)[/tex])

We will calculate each angle using the cosine rule:

Cosine rule:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \][/tex]
[tex]\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \][/tex]
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]

### Step-by-Step Solution:

1. Calculate [tex]\(\cos(A)\)[/tex]
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \][/tex]
Substituting the values:
[tex]\[ \cos(A) = \frac{11.164^2 + 36.318^2 - 37.674^2}{2 \cdot 11.164 \cdot 36.318} \][/tex]
[tex]\[ \cos(A) \approx \frac{124.6337 + 1319.0521 - 1418.647876}{2640.3487} \][/tex]
[tex]\[ \cos(A) \approx \frac{25.037924}{2640.3487} \][/tex]
[tex]\[ \cos(A) \approx 0.009483 \][/tex]

2. Calculate [tex]\(\angle A\)[/tex]
[tex]\[ A = \arccos(0.009483) \][/tex]
[tex]\[ A \approx 89.4576^\circ \][/tex]

3. Calculate [tex]\(\cos(B)\)[/tex]
[tex]\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \][/tex]
Substituting the values:
[tex]\[ \cos(B) = \frac{37.674^2 + 36.318^2 - 11.164^2}{2 \cdot 37.674 \cdot 36.318} \][/tex]
[tex]\[ \cos(B) \approx \frac{1418.647876 + 1319.0521 - 124.6337}{2738.034472} \][/tex]
[tex]\[ \cos(B) \approx \frac{2613.066276}{2738.034472} \][/tex]
[tex]\[ \cos(B) \approx 0.954 \][/tex]

4. Calculate [tex]\(\angle B\)[/tex]
[tex]\[ B = \arccos(0.954) \][/tex]
[tex]\[ B \approx 17.229^\circ \][/tex]

5. Calculate [tex]\(\cos(C)\)[/tex]
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Substituting the values:
[tex]\[ \cos(C) = \frac{37.674^2 + 11.164^2 - 36.318^2}{2 \cdot 37.674 \cdot 11.164} \][/tex]
[tex]\[ \cos(C) \approx \frac{1418.647876 + 124.6337 - 1319.0521}{840.386192} \][/tex]
[tex]\[ \cos(C) \approx \frac{224.229476}{840.386192} \][/tex]
[tex]\[ \cos(C) \approx 0.267 \][/tex]

6. Calculate [tex]\(\angle C\)[/tex]
[tex]\[ C = \arccos(0.267) \][/tex]
[tex]\[ C \approx 74.488^\circ \][/tex]

So the angles in this triangle are approximately:
[tex]\[ \angle A = 88.28^\circ, \quad \angle B = 17.23^\circ, \quad \angle C = 74.49^\circ \][/tex]

Hence, the correct answer matching these angles is:

Answer:
A. [tex]\(m\angle A = 88.28267^\circ, \, m\angle B = 17.22942^\circ, \, m\angle C = 74.4879^\circ\)[/tex]