points a and b are separated by a lake. to find the distance between them, a surveyor locates a point c on land such than $ \angle cab = 51.5^{\circ} $. find the distance across the lake from a to b.

note: the triangle is not drawn to scale.

distance = $ \square $ yd

enter your answer as a number; your answer should be accurate to 2 decimal places.

question help:
worked example 1

Question

points a and b are separated by a lake. to find the distance between them, a surveyor locates a point c on land such than $ \angle cab = 51.5^{\circ} $. find the distance across the lake from a to b.

note: the triangle is not drawn to scale.

distance = $ \square $ yd

enter your answer as a number; your answer should be accurate to 2 decimal places.

question help:
worked example 1

Accepted Answer

# Explanation:
## Step1: Apply the Law of Cosines
The Law of Cosines formula is $ c^{2}=a^{2}+b^{2}-2ab\cos C$. In $\triangle ABC$, let $a = 451$, $b=344$, and $C = 51.5^{\circ}$. We want to find the length of $AB$ (let's call it $c$).
## Step2: Substitute values into the formula
$$
\begin{align*}
c^{2}&=451^{2}+344^{2}-2\times451\times344\times\cos(51.5^{\circ})\
c^{2}&=203401 + 118336-311288\times\cos(51.5^{\circ})\
\end{align*}
$$
We know that $\cos(51.5^{\circ})\approx0.623$.
$$
\begin{align*}
c^{2}&=203401+118336-(311288\times0.623)\
c^{2}&=321737 - 194932.424\
c^{2}&=126804.576\
c&=\sqrt{126804.576}\
c&\approx356.10
\end{align*}
$$

# Answer:
$356.10$

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QUESTION IMAGE

Question

Explanation:

Step1: Apply the Law of Cosines

Step2: Substitute values into the formula

Answer: