A ship leaves its home port and sails on a bearing N38°15'E at 24 mph.At the same instant,another ship leaves the same port on a bearing S51°45'E at 28 mph.Find the distance between the two ships after 8 hours.
You have a triangle with sides 8(24) km and 8(28) km, and the contained angle of
(180 - 38°15; - 51°45') between them
Just use the cosine law
PAKYU
To find the distance between two ships after 8 hours, we need to calculate the distance each ship has traveled during this time.
For Ship 1:
Speed = 24 mph
Time = 8 hours
Distance = Speed * Time
Distance of Ship 1 = 24 mph * 8 hours
Distance of Ship 1 = 192 miles
For Ship 2:
Speed = 28 mph
Time = 8 hours
Distance = Speed * Time
Distance of Ship 2 = 28 mph * 8 hours
Distance of Ship 2 = 224 miles
Now, we can use the distance and direction information to find the distance between the two ships.
We can split Ship 1's path into two components, one in the north direction and the other in the east direction. The north component is given by the formula:
North Component = Distance of Ship 1 * cos(bearing)
North Component = 192 miles * cos(38°15')
Using a calculator, we find that cos(38°15') ≈ 0.801
North Component ≈ 192 miles * 0.801
North Component ≈ 153.792 miles
Similarly, we can calculate the east component of Ship 1's path using the formula:
East Component = Distance of Ship 1 * sin(bearing)
East Component = 192 miles * sin(38°15')
Using a calculator, we find that sin(38°15') ≈ 0.599
East Component ≈ 192 miles * 0.599
East Component ≈ 115.008 miles
Next, we can split Ship 2's path into two components using the same process.
South Component = Distance of Ship 2 * cos(bearing)
South Component = 224 miles * cos(51°45')
Using a calculator, we find that cos(51°45') ≈ 0.620
South Component ≈ 224 miles * 0.620
South Component ≈ 139.28 miles
East Component = Distance of Ship 2 * sin(bearing)
East Component = 224 miles * sin(51°45')
Using a calculator, we find that sin(51°45') ≈ 0.785
East Component ≈ 224 miles * 0.785
East Component ≈ 175.84 miles
Now, we can use these components to find the distance between the two ships using the Pythagorean theorem.
Distance = √((North Component - South Component)^2 + (East Component - West Component)^2)
Distance = √((153.792 - 139.28)^2 + (115.008 - 175.84)^2)
Using a calculator, we find that Distance ≈ √(196.168^2 + (-60.832)^2)
Distance ≈ √(38480.864 + 3698.542)
Distance ≈ √(42179.406)
Distance ≈ 205.37 miles
Therefore, the distance between the two ships after 8 hours is approximately 205.37 miles.
To find the distance between the two ships after 8 hours, we can break down the problem into two components: the horizontal distance (east-west) and the vertical distance (north-south).
For the first ship, which is traveling on a bearing of N38°15'E, we need to find the distance it has traveled in the north direction after 8 hours. The ship's speed is given as 24 mph, so the distance traveled in the north direction is:
Distance_North = Speed * Time = 24 mph * 8 hours
For the second ship, which is traveling on a bearing of S51°45'E, we need to find the distance it has traveled in the south direction after 8 hours. Since the ship is traveling south, the distance will be negative. Again, using the speed and time, we have:
Distance_South = - (Speed * Time) = - (28 mph * 8 hours)
To calculate the horizontal distance (east-west) traveled by both ships, we need to find the distance each ship has traveled in the east direction after 8 hours. The formula for the horizontal distance is:
Distance_East = Speed * Time = 24 mph * 8 hours = 192 miles
Now that we have calculated the distances in the north, south, and east directions, we can use the Pythagorean theorem to find the total distance between the two ships. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b):
c^2 = a^2 + b^2
In this case, a represents the distance traveled by the first ship in the north direction, and b represents the distance traveled by the second ship in the south direction. Therefore:
Total Distance^2 = Distance_North^2 + Distance_South^2 + Distance_East^2
Substituting the values we calculated earlier:
Total Distance^2 = (24 mph * 8 hours)^2 + (- (28 mph * 8 hours))^2 + (192 miles)^2
After calculating the total distance squared, you can take the square root to find the actual total distance between the two ships after 8 hours.
Note: Make sure to convert units consistently throughout the calculations.