Question

a boat traveled north for 28 miles, then turned x° southwest and traveled for 25 miles before stopping. when it stopped, the boat was 18 miles from its starting point. by how many degrees did the direction of the boat change when it made its first turn? round to the nearest degree. 30 degrees 39 degrees 46 degrees 50 degrees law of cosines: a² = b² + c² - 2bc cos(a)

Explanation:

Step1: Identify values for law of cosines

Let $a = 18$, $b = 28$, $c = 25$. We want to find the angle opposite side $a$, which is $x$. The law - of - cosines formula is $a^{2}=b^{2}+c^{2}-2bc\cos(A)$.

Step2: Rearrange the formula for $\cos(A)$

$\cos(A)=\frac{b^{2}+c^{2}-a^{2}}{2bc}$. Substitute $a = 18$, $b = 28$, $c = 25$ into the formula:
\[

$$\begin{align*} \cos(A)&=\frac{28^{2}+25^{2}-18^{2}}{2\times28\times25}\\ &=\frac{784 + 625-324}{1400}\\ &=\frac{1085}{1400}\\ & = 0.775 \end{align*}$$

\]

Step3: Find the angle

$A=\cos^{-1}(0.775)$. Using a calculator, $A\approx39^{\circ}$.

Answer:

39 degrees