A ferryboat transports tourists among three islands. It sails from the first island to the second island, 4.38 km away, in a direction 37.0° north of east. It then sails from the second island to the third island in a direction 76.5° west of north. Finally, it returns to the first island, sailing in a direction 28.0° east of south.
(a) Calculate the distance between the second and third islands.
(b) Calculate the distance between the first and third islands.
Oh, I do hope you mean "Physics" (notice the -s)!
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To calculate the distance between two points given their coordinates, we can use the formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Where (x1, y1) and (x2, y2) are the coordinates of the two points.
To find the distance between the second and third islands, we need to determine the coordinates of these points.
(a) Distance between the second and third islands:
1. Let's assume that the first island is the origin, (0, 0).
2. Let the second island be located at coordinates (x, y).
3. To find these coordinates, we will use the given information. The ferryboat sails from the first island to the second island in a direction 37.0° north of east. This means it is moving in a direction 37.0° above the positive x-axis.
- The x-component of the displacement is given by 𝑥_2−0=4.38 km×cos37.0°.
- The y-component of the displacement is given by 𝑦_2−0=4.38 km×sin37.0°.
4. Thus, the coordinates of the second island are (4.38 km × cos(37.0°), 4.38 km × sin(37.0°)).
Next, let's find the coordinates of the third island.
5. The ferryboat sails from the second island to the third island in a direction 76.5° west of north. This means it is moving in a direction 76.5° below the positive y-axis.
- The x-component of the displacement from the second island to the third island is given by 𝑥_3−𝑥_2=??? km × cos(76.5°).
- The y-component of the displacement from the second island to the third island is given by 𝑦_3−𝑦_2=??? km × sin(76.5°).
6. Thus, the coordinates of the third island are (4.38 km × cos(37.0°) + ??? km × cos(76.5°), 4.38 km × sin(37.0°) + ??? km × sin(76.5°)).
Finally, we can calculate the distance between the second and third islands using the formula for distance:
Distance = √((x3 - x2)^2 + (y3 - y2)^2)
(b) Distance between the first and third islands:
To find the distance between the first and third islands, we need to calculate the distance between the origin (first island) and the third island using the same formula for distance:
Distance = √((x3 - x1)^2 + (y3 - y1)^2), where (x1, y1) = (0, 0).
By performing these calculations, we will obtain the distances between the second and third islands as well as the first and third islands.