Chapter 6.6, Problem 39E

Surveying To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information.

To determine

To find: The distance across the small lake.

Answer to Problem 39E

The distance across the lake is $x\approx 2.30\text{mi}$.

Explanation of Solution

Given:

The below figure shows the given triangle $\Delta ABC$.

Figure (1)

Formula used: Law of Cosines

$c^2=a^2+b^2-2ab\cos C$.

Calculation:

Substitute $2.82$ for $a$, $3.56$ for $b$, $AB$ for $c$ and $40.3°$ for $C$ in law of cosines,

$\begin{array}{c}A{B}^2={\left(2.82\right)}^2+{\left(3.56\right)}^2-2\left(2.82\right)\left(3.56\right)\cos 40.3°\\ =7.9524+12.6736-20.0784\left(0.76267\right)\\ =20.626-15.31316\\ =5.31284\end{array}$

Simplify the above expression by taking square root to each side,

$\begin{array}{c}\sqrt{A{B}^2}=\sqrt{5.31284}\\ AB=2.30496\\ AB\approx 2.30\end{array}$

Thus, the distance across the lake is $x\approx 2.30\text{mi}$.