A ship leaves its home port expecting to travel to a port 500.0 km due south. Before it moves even 1 km, a servere storm blows it 100.0 km due east. Hom far is the ship from its destination? In what direction must it travel to reach its destination?
it must travel along the hypotenuse of a right triangle with sides 500 m (South) and 100 km (East)
Try using the Pythagorean theorem. Drawing yourself a sketch should help.
Show your work for furthe assistance, if needed.
To find out how far the ship is from its destination, we can use the Pythagorean theorem.
Let's call the distance traveled south as "y" and the distance traveled east as "x".
According to the question, the ship was originally planning to travel 500.0 km due south. However, due to the storm, it got blown 100.0 km due east before traveling any distance south.
Using the Pythagorean theorem, we can calculate the distance from the ship's starting point to its current position:
Distance^2 = y^2 + x^2
Distance^2 = (500.0 km - y)^2 + (100.0 km - x)^2
Since the ship has not traveled any distance south yet, y = 0.0 km.
Distance^2 = (500.0 km - 0.0 km)^2 + (100.0 km - x)^2
Distance^2 = (500.0 km)^2 + (100.0 km - x)^2
Now if we substitute this distance into the Pythagorean theorem, we can find the distance from the ship's destination:
Distance^2 = (500.0 km)^2 + (100.0 km - x)^2
Simplifying further:
Distance^2 = 250000.0 km^2 + (10000.0 km^2 - 200.0 kmx + x^2)
Expanding:
Distance^2 = 250000.0 km^2 + 10000.0 km^2 - 200.0 kmx + x^2
Distance^2 = 260000.0 km^2 - 200.0 kmx + x^2
Since the distance traveled south is 0 km, the ship's current position is (x, 0).
Now we can use this expression to calculate the distance:
Distance = sqrt(260000.0 km^2 - 200.0 kmx + x^2)
To determine the direction the ship needs to travel to reach its destination, we can use trigonometry.
The angle between the line connecting the starting point and the destination and the east direction can be found using the tangent function:
tan(theta) = opposite/adjacent
tan(theta) = y/x
tan(theta) = 0.0 km/x
tan(theta) = 0
This tells us that the angle theta between the line connecting the starting point and the destination and the east direction is 0 degrees.
Therefore, the ship needs to travel due south to reach its destination.
To find the ship's distance from its destination, we can use the Pythagorean theorem as the ship has essentially moved in a right-angled triangle.
Step 1: Determine the distance traveled horizontally (east) and vertically (south) using the given information.
- The ship was blown 100.0 km due east.
- The ship's original destination was 500.0 km due south.
- Therefore, the horizontal distance traveled is 100.0 km, and the vertical distance is 500.0 km.
Step 2: Apply the Pythagorean theorem to calculate the distance from the ship to its destination.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
In this case, let's consider the horizontal distance as the base (b) and the vertical distance as the height (h). The hypotenuse (d) is the distance from the ship to its destination.
Using the Pythagorean theorem, we have:
d² = b² + h²
d² = 100.0 km² + 500.0 km²
d² = 10,000 km² + 250,000 km²
d² = 260,000 km²
Taking the square root of both sides, we find:
d = √260,000 km
d ≈ 509.90 km
Therefore, the ship is approximately 509.90 km from its destination.
Step 3: Determine the direction the ship must travel to reach its destination.
Since the ship is 509.90 km away from its destination, and it has initially moved 100.0 km due east, we need to find the angle between the ship's location and its destination. This angle can be found using trigonometry.
Using the tangent function:
tan(angle) = h/b
tan(angle) = 500.0 km / 100.0 km
tan(angle) = 5
Taking the inverse tangent (arctan) of both sides:
angle = arctan(5)
angle ≈ 78.69°
Therefore, the ship must travel approximately 78.69° south of due east to reach its destination.