Chapter 5.6, Problem 49E

Solution

To find:The correct option from the given options.

The correct option is (C).

Given information:

Boat 1: Speed = $24\text{mph}$

Boat 2: Speed = $32\text{mph}$

The included angle in the paths of the two boats is $110°$ .

Calculation:

Let $x$ be the distance between the two boats after $30\min$ .

  

The formula to calculate distance is given by,

  $\text{Distance}\text{=}\text{Speed}\text{×}\text{Time}$

Distance travelled by Boat 1 in $30\min$ is

  $\begin{array}{c}\text{Distance}\left(a\right)=24\times 0.5\\ =12\text{miles}\end{array}$

Distance travelled by Boat 2 in $30\min$ is

  $\begin{array}{c}\text{Distance}\left(b\right)=32\times 0.5\\ =16\text{miles}\end{array}$

The formula for Law of Cosines is:

  $x^2=a^2+b^2-2ab\cos \theta$

Substitute $12$ for $a$, $16$ for $b$ and $110°$ for $\theta$ in $x^2=a^2+b^2-2ab\cos \theta$ to calculate the distance between the two boats after $30\min$ .

  $\begin{array}{c}{x}^2=a^2+b^2-2ab\cos \theta \\ =\sqrt{{12}^2+{16}^2-2\times 12\times 16\cos 110°}\\ \approx 23\text{miles}\end{array}$

Distance between the two boats after $30\min$ is $23\text{miles}$ .

Therefore, the correct option is (C).