Christopher wants to determine how far away a helicopter is from where he is standing. He knows the helicopter is sitting on the edge of a roof of an apartment building that is 400 feet high and 1,050 feet from where he is standing.
the distance is the hypotenuse of a triangle with legs 400 and 1050:
√(400^2 + 1050^2) = 1123 ft
To determine how far away the helicopter is from Christopher, we can use trigonometry. Let's consider the situation:
Christopher is standing on the ground, and the helicopter is sitting on the edge of a roof. We can form a right triangle with the helicopter at the top, Christopher at the bottom, and a line from the helicopter to Christopher forming the hypotenuse of the triangle.
The height of the building (400 feet) represents the opposite side of the triangle, and the distance between Christopher and the building (1,050 feet) represents the adjacent side of the triangle.
We can use the tangent function to calculate the angle of elevation (θ) between the line of sight from Christopher to the helicopter and the horizontal line on which he stands:
tan(θ) = opposite / adjacent
tan(θ) = 400 / 1050
To find the value of θ, we can take the inverse tangent (arctan) of both sides:
θ = arctan(400 / 1050)
Using a calculator or a trigonometry table, we find that θ ≈ 20.69 degrees.
Now, to determine the distance (x) between Christopher and the helicopter, we can use the sine function:
sin(θ) = opposite / hypotenuse
sin(θ) = 400 / x
Rearranging the equation, we get:
x = 400 / sin(θ)
Plugging in the value of θ, we can calculate:
x = 400 / sin(20.69)
Using a calculator, we find that x ≈ 1,117.11 feet.
Therefore, the helicopter is approximately 1,117.11 feet away from where Christopher is standing.