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.
To calculate the distance between the second and third islands, we can use the law of cosines. Let's denote the distance between the second and third islands as d23.
(a) Distance between the second and third islands (d23):
Using the law of cosines,
c^2 = a^2 + b^2 - 2ab*cos(C)
where c is the side opposite to angle C.
Given:
a = 4.38 km (distance between the first and second islands)
B = 76.5° (angle between the direction from the second island to the third island and the north direction)
We want to find c, which is the distance between the second and third islands.
Therefore, a = d23, b = 4.38 km, and C = 76.5°
Plugging in the values into the formula,
d23^2 = (4.38)^2 + (4.38)^2 - 2 * 4.38 * 4.38 * cos(76.5°)
d23^2 ≈ 38.3044 + 38.3044 - 2 * 4.38 * 4.38 * 0.2356
d23^2 ≈ 76.6088 - 20.5924232
d23^2 ≈ 56.0163768
Taking the square root of both sides,
d23 ≈ √56.0163768
d23 ≈ 7.483 km
Therefore, the distance between the second and third islands is approximately 7.483 km.
(b) Distance between the first and third islands:
To calculate the distance between the first and third islands, we need to add the distances between the first and second islands and between the second and third islands.
Given:
a = 4.38 km (distance between the first and second islands).
d23 ≈ 7.483 km (distance between the second and third islands).
Adding the distances,
d13 = a + d23
d13 = 4.38 km + 7.483 km
d13 ≈ 11.863 km
Therefore, the distance between the first and third islands is approximately 11.863 km.
To solve this problem, we can break it down into smaller parts and use vector components to find the distances between the islands.
(a) Calculate the distance between the second and third islands:
Step 1: Determine the x and y components of each leg of the journey.
First leg (from first island to second island):
Distance = 4.38 km
Direction = 37.0° north of east
To find the x-component:
x1 = 4.38 km * cos(37.0°)
To find the y-component:
y1 = 4.38 km * sin(37.0°)
Second leg (from second island to third island):
Distance = ?
Direction = 76.5° west of north
To find the x-component:
x2 = ? km * cos(76.5°)
To find the y-component:
y2 = ? km * sin(76.5°)
Last leg (from third island to the first island):
Distance = ?
Direction = 28.0° east of south
To find the x-component:
x3 = ? km * cos(28.0°)
To find the y-component:
y3 = ? km * sin(28.0°)
Step 2: Determine the net x and y components.
Net x-component = x1 + x2 + x3
Net y-component = y1 + y2 + y3
Step 3: Use the net x and y components to find the net distance (distance between second and third islands).
Distance = sqrt((Net x-component)^2 + (Net y-component)^2)
(b) Calculate the distance between the first and third islands:
Step 1: Determine the x and y components of each leg of the journey.
First leg (from first island to second island):
Distance = 4.38 km
Direction = 37.0° north of east
To find the x-component:
x1 = 4.38 km * cos(37.0°)
To find the y-component:
y1 = 4.38 km * sin(37.0°)
Second leg (from second island to third island):
Distance = ?
Direction = 76.5° west of north
To find the x-component:
x2 = ? km * cos(76.5°)
To find the y-component:
y2 = ? km * sin(76.5°)
Step 2: Determine the net x and y components.
Net x-component = x1 + x2
Net y-component = y1 + y2
Step 3: Use the net x and y components to find the net distance (distance between first and third islands).
Distance = sqrt((Net x-component)^2 + (Net y-component)^2)
Note: To find the missing distances and directions for the second and third legs, additional information is needed.