solving a navigation problem
a ferry transports passengers to three different ports. from port a, the ferry travels east 3.5 miles to port b. the ship then turns 75° south and travels for 6 miles to reach port c. what is the distance d from port c to port a? approximate to the nearest tenth of a mile.
□ miles

Question

Accepted Answer

# Explanation:
## Step1: Determine the angle at B
The angle at $ B$ in $\triangle ABC$ is $180^{\circ}- 75^{\circ}=105^{\circ}$.
## Step2: Apply the Law of Cosines
The Law of Cosines states that $d^{2}=AB^{2}+BC^{2}-2(AB)(BC)\cos B$.
Given $AB = 3.5$, $BC = 6$, and $B=105^{\circ}$, we substitute these values into the formula:
$$
\begin{align*}
d^{2}&=3.5^{2}+6^{2}-2\times3.5\times6\times\cos(105^{\circ})\
d^{2}&=12.25 + 36-42\times(-0.2588)\
d^{2}&=48.25+10.87\
d^{2}&=59.12\
d&=\sqrt{59.12}\
d&\approx7.7
\end{align*}
$$

# Answer:
$7.7$ miles

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QUESTION IMAGE

Question

Explanation:

Step1: Determine the angle at B

Step2: Apply the Law of Cosines

Answer: