1 boat travels east at 12 mph. The other boat travels northeast at 16 mph. How far apart will they be after 2 hours?

Question

The boats travel on the same course the entire time.

Answer 1

This is a problem of solving a triangle using the law of cosines: the square of measure of the unknown side is equal to the sum of the squares of the other two sides minus 2 times the product of those sides times the cosine of the angle opposite the unknown side. That the boats are traveling along their same courses the entire time means they travel in straight lines. Now, think about the compass. North & east make a 90 degree angle, while northeast is exactly between them - 45 degrees. So, we have two straight lines that form a 45 degree angle, and we will use this information to measure the length of the unknown line between the two boats after two hours.

 

The eastern boat, traveling at 12 mph, covers 24 miles in 2 hours, while the northeastern boat, at 16 mph, travels 32 miles during that same period. Now we make our equation & solve for a:

 

a² = b² + c² - 2bc(cos(45°))

a² = (24)² + (32)² - 2(24)(32)(0.70711)

a² = 576 + 1024 - 2(768)(0.70711)

a² = 1600 - 1536(0.70711)

a² = 1600 - 1086.116

a² = 513.884

a = 22.669 miles

Answer 2

First off, we need to figure out where the boats are. Boat A at 12MPH after 2 hours means it is at (24,0) and Boat B at 16 MPH make a 45 degree right triangle with a hypotenuse of 32. A leg is proportional to the hypotenuse at a ratio of x:x√2. That means 32/√2=X. So the location of Boat B after 2 hours is (32/√2,32/√2)

The formula for the distance between two points, in this case the boats, is

D=√[(X₂- X₁)²+(Y₂- Y₁)²]

D=√[(32/√2-24)²+(32/√2-0)²]=√[(512+(16√2-24)²]=22.669009...≅22.67

They are approximately 22.67 Miles apart after two hours