three boats are at sea: voyager (v), escape(e), and freedom(f). the voyager can see e and f. the angle between the line of sight to e and the line of sight to f is 45 degrees. if the distance between v and e is 2 miles and the distance between v and f is 4 miles, what is the distance between e & f?

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Accepted Answer

# Explanation:
## Step1: Use the Law of Cosines
The Law of Cosines formula is $EF^{2}=VE^{2}+VF^{2}-2\times VE\times VF\times\cos\angle EVF$.
Let $VE = a = 2$, $VF = b = 4$, $\angle EVF=\theta = 45^{\circ}$, and $EF = c$.
## Step2: Substitute values into the formula
Substitute $a = 2$, $b = 4$, $\theta = 45^{\circ}$ ($\cos45^{\circ}=\frac{\sqrt{2}}{2}$) into the formula:
$$
\begin{align*}
c^{2}&=2^{2}+4^{2}-2\times2\times4\times\frac{\sqrt{2}}{2}\
c^{2}&=4 + 16-8\sqrt{2}\
c^{2}&=20-8\sqrt{2}\
c^{2}&\approx20 - 8\times1.414\
c^{2}&\approx20-11.312\
c^{2}&\approx8.688\
c&\approx2.95
\end{align*}
$$

# Answer:
The distance between $E$ and $F$ is approximately $2.95$ miles.

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QUESTION IMAGE

Question

Explanation:

Step1: Use the Law of Cosines

Step2: Substitute values into the formula

Answer: