Question
Two ships leave the same port in different directions, forming a 120° angle between them. One ship travels 70 mi. and the other 52 mi. before they each drop their anchors. What is the distance between the ships to the nearest mile?
(1 point)
Responses
87 mi.
87 mi.
98 mi.
98 mi.
106 mi.
106 mi.
122 mi.
(1 point)
Responses
87 mi.
87 mi.
98 mi.
98 mi.
106 mi.
106 mi.
122 mi.
Answers
GPT-4o mini
To find the distance between the two ships after they have traveled in different directions, we can use the Law of Cosines.
The formula for the Law of Cosines is:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta) \]
where:
- \( c \) is the side opposite the angle \( \theta \),
- \( a \) and \( b \) are the lengths of the other two sides,
- \( \theta \) is the angle between sides \( a \) and \( b \).
In this case:
- \( a = 70 \) mi (distance traveled by the first ship),
- \( b = 52 \) mi (distance traveled by the second ship),
- \( \theta = 120^\circ \).
First, we need to calculate \( \cos(120^\circ) \):
\[
\cos(120^\circ) = -\frac{1}{2}
\]
Now, we can plug the values into the Law of Cosines formula:
\[
c^2 = 70^2 + 52^2 - 2 \cdot 70 \cdot 52 \cdot \left(-\frac{1}{2}\right)
\]
Calculating the squares:
\[
70^2 = 4900, \quad 52^2 = 2704
\]
Calculating the product:
\[
2 \cdot 70 \cdot 52 \cdot \left(-\frac{1}{2}\right) = -70 \cdot 52 = -3640
\]
Now substituting these values into the equation:
\[
c^2 = 4900 + 2704 + 3640
\]
\[
c^2 = 4900 + 2704 + 3640 = 11244
\]
Now, take the square root to find \( c \):
\[
c = \sqrt{11244} \approx 106.02
\]
Rounding to the nearest mile gives \( c \approx 106 \) mi.
Thus, the answer is:
**106 mi.**
The formula for the Law of Cosines is:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta) \]
where:
- \( c \) is the side opposite the angle \( \theta \),
- \( a \) and \( b \) are the lengths of the other two sides,
- \( \theta \) is the angle between sides \( a \) and \( b \).
In this case:
- \( a = 70 \) mi (distance traveled by the first ship),
- \( b = 52 \) mi (distance traveled by the second ship),
- \( \theta = 120^\circ \).
First, we need to calculate \( \cos(120^\circ) \):
\[
\cos(120^\circ) = -\frac{1}{2}
\]
Now, we can plug the values into the Law of Cosines formula:
\[
c^2 = 70^2 + 52^2 - 2 \cdot 70 \cdot 52 \cdot \left(-\frac{1}{2}\right)
\]
Calculating the squares:
\[
70^2 = 4900, \quad 52^2 = 2704
\]
Calculating the product:
\[
2 \cdot 70 \cdot 52 \cdot \left(-\frac{1}{2}\right) = -70 \cdot 52 = -3640
\]
Now substituting these values into the equation:
\[
c^2 = 4900 + 2704 + 3640
\]
\[
c^2 = 4900 + 2704 + 3640 = 11244
\]
Now, take the square root to find \( c \):
\[
c = \sqrt{11244} \approx 106.02
\]
Rounding to the nearest mile gives \( c \approx 106 \) mi.
Thus, the answer is:
**106 mi.**