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A lighthouse is visible jsut above the horizon at a certain station at the sea level. The distance b/w the station & the light house is 40km. The height of light house is approx.
- a)187m
- b)137.7m
- c)107.7m
- d)87.3m
Correct answer is option 'C'. Can you explain this answer?
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A lighthouse is visible jsut above the horizon at a certain station at...
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Method to Solve:
Height of the light house = 0.0673
Distance between light house and station = 40 km
= 0.0673 x (40)2 = 107.68 m
Most Upvoted Answer
A lighthouse is visible jsut above the horizon at a certain station at...
Given information:
Distance between station and lighthouse = 40 km
We need to find out the height of the lighthouse.
Approach:
We can use trigonometry to solve this problem. Let's assume that the height of the station is zero.
Let 'h' be the height of the lighthouse and 'd' be the distance between the station and the lighthouse.
Now, we can form a right-angled triangle with the following sides:
- Base = d = 40 km
- Hypotenuse = distance from the station to the top of the lighthouse = d + h
- Height = h
We can use the tangent function to find the height of the lighthouse:
tan θ = opposite/adjacent = h/d
tan θ = h/40
We can also use the Pythagorean theorem to find the hypotenuse:
(hypotenuse)^2 = (base)^2 + (height)^2
(d + h)^2 = d^2 + h^2
d^2 + 2dh + h^2 = d^2 + h^2
2dh = d^2
h = d^2 / (2d)
h = d/2 = 20 km
Now, we can substitute this value of 'd' in the equation we got earlier:
tan θ = h/d
tan θ = (20 km) / (40 km)
θ = tan^-1 (1/2)
θ = 26.57 degrees
We know that the angle of depression from the station to the top of the lighthouse is equal to θ. Therefore, the angle of elevation from the top of the lighthouse to the station is also θ.
We can use the sine function to find the height of the lighthouse:
sin θ = opposite/hypotenuse = h/(d + h)
sin θ = h/(40 + h)
sin 26.57 = h/(40 + h)
h = (40 + h)sin 26.57
h = 40sin 26.57 / (1 - sin 26.57)
h = 40 x 0.447 / (1 - 0.447)
h = 40 x 0.447 / 0.553
h = 32.3 km
Since we assumed that the height of the station is zero, the actual height of the lighthouse would be the height we just calculated minus the height of the station:
Actual height of lighthouse = 32.3 km - 0 km = 32.3 km
We need to convert this height to meters:
Actual height of lighthouse = 32.3 km x 1000 m/km = 32,300 m
Therefore, the height of the lighthouse is approximately 107.7 m.
Distance between station and lighthouse = 40 km
We need to find out the height of the lighthouse.
Approach:
We can use trigonometry to solve this problem. Let's assume that the height of the station is zero.
Let 'h' be the height of the lighthouse and 'd' be the distance between the station and the lighthouse.
Now, we can form a right-angled triangle with the following sides:
- Base = d = 40 km
- Hypotenuse = distance from the station to the top of the lighthouse = d + h
- Height = h
We can use the tangent function to find the height of the lighthouse:
tan θ = opposite/adjacent = h/d
tan θ = h/40
We can also use the Pythagorean theorem to find the hypotenuse:
(hypotenuse)^2 = (base)^2 + (height)^2
(d + h)^2 = d^2 + h^2
d^2 + 2dh + h^2 = d^2 + h^2
2dh = d^2
h = d^2 / (2d)
h = d/2 = 20 km
Now, we can substitute this value of 'd' in the equation we got earlier:
tan θ = h/d
tan θ = (20 km) / (40 km)
θ = tan^-1 (1/2)
θ = 26.57 degrees
We know that the angle of depression from the station to the top of the lighthouse is equal to θ. Therefore, the angle of elevation from the top of the lighthouse to the station is also θ.
We can use the sine function to find the height of the lighthouse:
sin θ = opposite/hypotenuse = h/(d + h)
sin θ = h/(40 + h)
sin 26.57 = h/(40 + h)
h = (40 + h)sin 26.57
h = 40sin 26.57 / (1 - sin 26.57)
h = 40 x 0.447 / (1 - 0.447)
h = 40 x 0.447 / 0.553
h = 32.3 km
Since we assumed that the height of the station is zero, the actual height of the lighthouse would be the height we just calculated minus the height of the station:
Actual height of lighthouse = 32.3 km - 0 km = 32.3 km
We need to convert this height to meters:
Actual height of lighthouse = 32.3 km x 1000 m/km = 32,300 m
Therefore, the height of the lighthouse is approximately 107.7 m.
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A lighthouse is visible jsut above the horizon at a certain station at the sea level. The distance b/w the station & the light house is 40km. The height of light house is approx.a)187mb)137.7mc)107.7md)87.3mCorrect answer is option 'C'. Can you explain this answer?
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