a wire supporting a 50 ft radio antenna stretched from the top of the antenna to the ground 32 ft away from the base of the antenna. find the angle formed by the wire and the ground
To find the angle formed by the wire and the ground, we can draw a right triangle.
Let's call the length of the wire "w" and the distance from the base of the antenna to the ground "d".
In this case, w = 50 ft and d = 32 ft.
Using the Pythagorean theorem, we can find the length of the unknown side of the triangle (which is the height of the triangle).
Using the formula a^2 + b^2 = c^2, where a and b are the two shorter sides of the triangle and c is the hypotenuse, we have:
(32 ft)^2 + h^2 = (50 ft)^2
1024 ft^2 + h^2 = 2500 ft^2
h^2 = 2500 ft^2 - 1024 ft^2
h^2 = 1476 ft^2
h = √1476 ft
h ≈ 38.39 ft
Now, let's find the sine of the angle formed by the wire and the ground.
Sine (θ) = opposite/hypotenuse
Sine (θ) = h/w
Sine (θ) = 38.39 ft/50 ft
To find the angle, we need to find the inverse sine (sine^-1) or arcsin of this value.
θ = arcsin(38.39 ft/50 ft)
Using a calculator or a trigonometric table, we find:
θ ≈ 52.43 degrees
Therefore, the angle formed by the wire and the ground is approximately 52.43 degrees.