Two boats start out from the same dock traveling at 9 mph and 12 mph respectively. The boats travel in slightly different directions and after 20 minutes are 2 miles apart. What is the angle that is formed by the paths of the two boats? I don't know how to set this up please help and explain!
20 min = 1/3 hour so
boat a goes 3 miles and boat b goes 4 miles
sides of triangle are 2, 3 and 4 miles
use law of cosines to find angle opposite the 2 mile side.
Thank you so much!
To find the angle formed by the paths of the two boats, we can use some trigonometry. Let's break down the problem into steps and explain each step along the way:
Step 1: Set up a diagram
Start by drawing a diagram to visualize the scenario. Draw a line to represent the distance between the two boats after 20 minutes. The two boats start at the same point, but they travel in slightly different directions.
Step 2: Find the distances covered by each boat in 20 minutes
We know that the speed of the first boat is 9 mph and the speed of the second boat is 12 mph. Since both boats have traveled for 20 minutes, which is 20/60 = 1/3 hour, multiply the speed of each boat by 1/3 to find the distance covered by each boat. The first boat has covered (9 mph) × (1/3 hour) = 3 miles, and the second boat has covered (12 mph) × (1/3 hour) = 4 miles.
Step 3: Use the Law of Cosines to find the angle
Now that we have the distances covered by each boat, we can calculate the angle. The Law of Cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those two sides and the cosine of the included angle.
In our case, the sides of the triangle are the distances covered by the two boats, which are 3 miles and 4 miles, and the included angle is the angle we want to find.
The Law of Cosines equation is:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the side opposite the angle we want to find (the distance of 2 miles between the boats), and a, b, and C are the sides and angle we know.
Plugging in the values we have:
2^2 = 3^2 + 4^2 - 2(3)(4) * cos(C)
Simplifying the equation:
4 = 9 + 16 - 24 * cos(C)
4 = 25 - 24 * cos(C)
Solving for cos(C):
20 * cos(C) = 21
cos(C) = 21/20
Step 4: Determine the angle
Now that we have the value of cos(C), we can find the angle by taking the inverse cosine (cos^{-1}) of cos(C) using a calculator.
cos^{-1}(21/20) ≈ 11.47°
So, the angle formed by the paths of the two boats is approximately 11.47°.