A tower 125 feet high stands on the side of a hill. At a point 240 feet from the foot of the tower measured straight down the hill, the tower subtends an angle of 25 degrees. What angle does the side of the hill make with the horizontal?
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To find the angle that the side of the hill makes with the horizontal, we can use trigonometry.
Let's denote the angle we are trying to find as θ.
First, let's draw a diagram representing the situation. We have a tower that is 125 feet high and a point on the hill that is 240 feet away from the foot of the tower. The angle between the line connecting the tower's foot and the point on the hill is 25 degrees. We want to find the angle that the side of the hill makes with the horizontal.
Now, let's consider a right-angled triangle formed by the tower, the point on the hill, and the foot of the tower. The side of the triangle opposite angle θ represents the height of the hill, and the side adjacent to angle θ represents the distance along the horizontal.
We can use the tangent function to find θ. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side of a right-angled triangle.
Therefore, we have:
tan(θ) = opposite / adjacent,
tan(θ) = 125 / 240.
To find θ, take the inverse tangent (arctan) of both sides:
θ = arctan(tan(125 / 240)).
Use a calculator to find the inverse tangent of the tangent of 125 / 240. The resulting value will be the angle θ.
So, to calculate the angle that the side of the hill makes with the horizontal, type "arctan(tan(125 / 240))" in a calculator, and it will give you the answer.