ou are a pirate captain trying to determine how long it will take you to get to an island where you expect to find buried treasure. You are heading due North, toward the island. 60° West of North, you spot a light house. According to your sea chart, the top of the lighthouse is 50 m off the ground. You can see the top of the light house at a 5° incline to the horizon. The sea chart also tells you that the distance from the lighthouse to the island is 512.39 m. If the boat is travelling at 5 m/s, how long will it take for the boat to reach the island?
Draw a diagram. Let
T be the top of the lighthouse
B be the bottom of the lighthouse
S be the location of the ship
I be the location of the island
Then SB, the distance to the lighthouse can be found using
50/SB = tan5°
SB = 571.50 m
Now, angle θ=BIS can be found using
sinθ/571.50 = sin60°/512.39
sinθ = 0.9659
θ = 75°
That means that angle SBI = 45°
So, the distance SI to the island is
SI^2 = 572.39^2 + 571.50^2 - 2*572.39*571.50*cos45° = 191624
SI = 437.75 m
So, at 5m/s, it will take 437.75/5 = 87.55 seconds to arrive at the island
To determine how long it will take for the boat to reach the island, we need to calculate the distance and then divide it by the boat's velocity.
First, let's find the distance between the boat and the lighthouse. We can use trigonometry to do this.
1. Find the vertical distance (height) from the boat to the lighthouse.
tan(5°) = height / distance
height = distance * tan(5°)
height = 512.39 m * tan(5°)
height ≈ 44.92 m
2. Find the horizontal distance from the boat to the lighthouse.
cos(5°) = horizontal distance / distance
horizontal distance = distance * cos(5°)
horizontal distance = 512.39 m * cos(5°)
horizontal distance ≈ 507.74 m
3. Since the boat is heading due North, we can use the Pythagorean theorem to find the distance between the boat and the lighthouse.
distance^2 = (horizontal distance)^2 + (vertical distance)^2
distance^2 ≈ (507.74 m)^2 + (44.92 m)^2
distance ≈ sqrt((507.74 m)^2 + (44.92 m)^2)
distance ≈ sqrt(257826.92 m^2 + 2016.45 m^2)
distance ≈ sqrt(259843.37 m^2)
distance ≈ 509.77 m
Now that we have the distance between the boat and the lighthouse, we can calculate how long it will take to reach the island.
4. Calculate the time it takes to reach the island.
time = distance / velocity
time = 509.77 m / 5 m/s
time ≈ 101.95 s
Therefore, it will take approximately 101.95 seconds for the boat to reach the island.
To determine how long it will take for the boat to reach the island, we need to find the distance the boat will travel.
First, let's find the height of the lighthouse. Given that the top of the lighthouse is 50 m off the ground and you can see it at a 5° incline to the horizon, we can use trigonometry to find the height.
The height of the lighthouse can be found using the equation: height = distance * tan(angle).
Given the angle (5°) and the distance to the lighthouse (512.39 m), let's calculate the height of the lighthouse:
height = 512.39 m * tan(5°) = 44.84 m
Now that we know the height of the lighthouse, let's determine the distance the boat will travel.
The boat is heading due North, and 60° West of North, the boat spots the lighthouse. This forms a right-angled triangle between the boat, the lighthouse, and the point where the boat is at the time of observation.
Using trigonometry, we can determine the distance the boat will travel:
distance = height / tan(60°)
Given the height (44.84 m) and the angle (60°), let's calculate the distance:
distance = 44.84 m / tan(60°) = 24.40 m
Now that we know the distance the boat will travel (24.40 m) and the speed of the boat (5 m/s), we can calculate the time it will take for the boat to reach the island:
time = distance / speed
Given the distance (24.40 m) and the speed (5 m/s), let's calculate the time:
time = 24.40 m / 5 m/s = 4.88 s
Therefore, it will take approximately 4.88 seconds for the boat to reach the island.