A sailboat heads north at 3 m/s for 1 hour and then tracks back to the southeast (at 45 degrees to north) at 2 m/s for 45 minutes.
a. How far has the boat sailed?
b. How far is it from its starting location?
The textbook that included the word problem is: Basic Biomechanics Sixth Edition by Susan J. Hall;
ISBN#978-0-07-337644-8
The course name: Biomechanics
http://www.jiskha.com/display.cgi?id=1380495585
Please help me
The boat's northbound distance = 3(3600) m = 10800 m or 10.8 km
then the distance going southeast = 2(45)(60) = 5400 m or 5.4 km
Did you make a sketch?
Mine is a triangle with sides 10800 and 5400 with a 45° angle between them
a) the total distance traveled is simply the sum of these two distances = 16200 m or 16.2 km
b) Looks ideally set up for the cosine law
let the distance from end point to starting point be x m
x^2 = 10800^2 + 5400^2 - 2(5400)(10800)cos45°
...
x = √.....
= 7957.6 m
or appr 7.96 km
check my arithmetic
a. Well, let's crunch some numbers here. In the first hour, the sailboat is heading north at 3 m/s, so it covers a distance of 3 meters per second for 1 hour, which is 3 x 60 x 60 = 10,800 meters.
In the next 45 minutes, the sailboat is heading southeast at 2 m/s, which is 2 x 45 x 60 = 5,400 meters.
So, the total distance sailed by the boat is 10,800 + 5,400 = 16,200 meters.
b. Now, to figure out how far the boat is from its starting location, we need to break it down to some technical stuff.
Since the boat initially travels north for 1 hour at 3 m/s, it covers a distance of 3 x 60 x 60 = 10,800 meters north.
Then, it sails southeast for 45 minutes at 2 m/s, which covers a distance of 2 x 45 x 60 = 5,400 meters southeast.
Now, let's whip out some trigonometry skills! Since the boat is sailing at a 45-degree angle to the north, we can use the cosine of 45 degrees to calculate the horizontal displacement:
Horizontal displacement = 5,400 meters x cos(45 degrees) = 5,400 x 0.707 ≈ 3,819 meters.
And to find the vertical displacement, we can use the sine of 45 degrees:
Vertical displacement = 10,800 meters x sin(45 degrees) = 10,800 x 0.707 ≈ 7,650 meters.
Finally, we can use the Pythagorean theorem to find the boat's distance from its starting location:
Distance = √(3,819^2 + 7,650^2) ≈ 8,535 meters.
So, the sailboat is approximately 8,535 meters away from its starting location. Just a little adventure off the beaten path! How nautical!
To find the answers to the questions, we can use basic trigonometry and the concept of displacement.
a. To calculate how far the boat has sailed, we need to find the total distance covered in each direction and then add them together.
The boat travels north for 1 hour at 3 m/s, so the distance traveled in the north direction is:
Distance north = Speed x Time = 3 m/s x 1 hour = 3 m/s x 3600 seconds = 10,800 meters.
Next, the boat tracks back to the southeast at 45 degrees to north for 45 minutes. To find the distance covered in this direction, we need to convert the time to hours:
Time in hours = 45 minutes / 60 minutes = 0.75 hours.
The distance covered in the southeast direction is:
Distance southeast = Speed x Time = 2 m/s x 0.75 hours = 1.5 m/s x 2,700 seconds = 4,050 meters.
Now we can find the total distance sailed by adding the two distances:
Total distance sailed = Distance north + Distance southeast = 10,800 meters + 4,050 meters = 14,850 meters.
Therefore, the boat has sailed a total distance of 14,850 meters.
b. To find how far the boat is from its starting location, we need to find the displacement. Displacement is the straight-line distance between the starting and ending points of the boat's journey.
To calculate the displacement, we can use vector addition. We know the distances covered in the north and southeast directions, but we need to find the resultant vector.
Using trigonometry, we can decompose the distance southeast into its north and east components. The southeast angle is 45 degrees.
North component = Distance southeast x cos(45 degrees) = 4,050 meters x cos(45 degrees) = 4,050 meters x 0.707 = 2,864.35 meters.
East component = Distance southeast x sin(45 degrees) = 4,050 meters x sin(45 degrees) = 4,050 meters x 0.707 = 2,864.35 meters.
Now we can find the displacement by subtracting the north and east components from the distance north (since the boat is moving in the opposite direction):
Displacement = Distance north - North component = 10,800 meters - 2,864.35 meters = 7,935.65 meters.
Therefore, the boat is approximately 7,935.65 meters away from its starting location.