Ship A and Ship B leave from the same point in the ocean. Ship A travels 150 mi due west, turns 65° toward north, and then travels another 100 mi. Ship B travels 150 mi due east, turns 70° toward south, and then travels another 100 mi. Which ship is farther from the starting point? Explain.
When you sketch the diagrams for ship A and B, the first one has sides of 150 and 100 with a contained angle of 115°.
Ship B's diagram has two sides 150 and 100 with a contained angle of 110°
the side opposite the smaller angle would be the shorter side.
So without any calculations, we know that ship A is farther from the starting point.
Since that is all that was asked for, we are done
wait would Ship A be the smaller angle(65) so would Ship B be the bigger length.
To determine which ship is farther from the starting point, we need to calculate the final positions of both Ship A and Ship B. Let's break down the steps for each ship:
Ship A:
1. Ship A travels 150 miles due west.
2. Ship A then turns 65° toward the north.
3. Ship A travels another 100 miles.
Ship B:
1. Ship B travels 150 miles due east.
2. Ship B then turns 70° toward the south.
3. Ship B travels another 100 miles.
Now let's determine the final positions of both ships:
For Ship A:
- The distance traveled due north after turning is given by 100 * sin(65°).
- The distance traveled due west is 150 miles.
- Therefore, the final north-south position is 100 * sin(65°) miles north, and the final east-west position is -150 miles.
For Ship B:
- The distance traveled due south after turning is given by 100 * sin(70°).
- The distance traveled due east is 150 miles.
- Therefore, the final north-south position is -100 * sin(70°) miles south, and the final east-west position is 150 miles.
To determine which ship is farther from the starting point, we need to calculate the total distance traveled by both ships.
For Ship A:
The total distance traveled by Ship A is the magnitude of the displacement vector from the starting point to the final position, which is given by the square root of (east-west position squared + north-south position squared).
For Ship B:
The total distance traveled by Ship B is calculated the same way.
By comparing the total distances traveled by Ship A and Ship B, we can determine which ship is farther from the starting point.
To determine which ship is farther from the starting point, we need to calculate the final positions of both Ship A and Ship B.
Let's break down the movements of both ships step by step:
1. Ship A travels 150 miles due west.
2. It then turns 65° toward the north and travels another 100 miles.
To find the final position of Ship A, we can use basic trigonometry. The 150-mile leg going due west is the adjacent side of a right triangle, while the 100-mile leg moving toward the north is the opposite side. Since we know the angle formed between the westward leg and the northward leg is 65°, we can use the tangent function:
tan(65°) = opposite / adjacent
tan(65°) = (100 miles) / (150 miles)
Now, we can solve for the value of the opposite side (100 miles). Rearranging the equation:
opposite = tan(65°) * adjacent
opposite = tan(65°) * 150 miles
By calculating this, we find that the northward distance for Ship A is approximately 187.3 miles. Consequently, the final position of Ship A can be identified as 187.3 miles north and 150 miles west.
Now, let's analyze Ship B's movements:
1. Ship B travels 150 miles due east.
2. It then turns 70° toward the south and travels another 100 miles.
Again, we can apply trigonometry to determine the final position of Ship B. In this case, we use the same method as before, but this time we calculate the southward (opposite) distance.
tan(70°) = opposite / adjacent
tan(70°) = (100 miles) / (150 miles)
Solving for the value of the opposite side:
opposite = tan(70°) * adjacent
opposite = tan(70°) * 150 miles
By calculating this, we find that the southward distance for Ship B is approximately 231.6 miles. Thus, the final position of Ship B can be identified as 231.6 miles south and 150 miles east.
To determine which ship is farther from the starting point, we can use the Pythagorean theorem. The hypotenuse of the right-angled triangle formed by Ship A's final position represents its distance from the starting point, and the same applies to Ship B.
For Ship A:
distance^2 = (187.3 miles)^2 + (150 miles)^2
For Ship B:
distance^2 = (231.6 miles)^2 + (150 miles)^2
By calculating the square of the sum of the squares for each ship, we find that the value for Ship A is approximately 32089.29 square miles, and the value for Ship B is approximately 38069.56 square miles.
Comparing these values, we can see that Ship B is farther from the starting point since its calculated distance is larger.
Therefore, Ship B is farther from the starting point than Ship A.