Question

a sailboat is 12 km north of a lighthouse. a motor cruiser is 12 km east of the same lighthouse. use trigonometry to find an exact expression for the distance between the two boats. your answer is:

Explanation:

Step1: Identify the right - triangle

The positions of the sailboat, motor - cruiser, and lighthouse form a right - triangle. The distance of the sailboat from the lighthouse $a = 12$ km (north direction, vertical side), the distance of the motor - cruiser from the lighthouse $b = 12$ km (east direction, horizontal side), and the distance between the two boats $d$ is the hypotenuse of the right - triangle.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with sides $a$, $b$, and hypotenuse $c$, $c^{2}=a^{2}+b^{2}$. Here, $d^{2}=12^{2}+12^{2}$.
$d^{2}=144 + 144=288$.

Step3: Solve for $d$

Take the square root of both sides to find $d$.
$d=\sqrt{288}=\sqrt{144\times2}=12\sqrt{2}$ km.

Answer:

$12\sqrt{2}$ km