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 the side of the hill makes with the horizontal, we can use the tangent function.

Let's call the angle the side of the hill makes with the horizontal "θ."

In a right triangle with the tower, the side of the hill, and the horizontal distance of 240 feet, we have:

Opposite side = 125 feet (height of the tower)
Adjacent side = 240 feet

We can express the tangent of θ as the ratio of the opposite side to the adjacent side:

tan(θ) = Opposite side / Adjacent side

Substituting the known values:

tan(θ) = 125 / 240

Next, we can find the value of θ by taking the inverse tangent (arctan) of both sides:

θ = arctan(125 / 240)

Using a calculator or trigonometric table, we can find the value of θ:

θ ≈ 27.02 degrees

Therefore, the angle the side of the hill makes with the horizontal is approximately 27.02 degrees.

To find the angle that the side of the hill makes with the horizontal, we can use trigonometry. Here's how you can calculate it step-by-step:

Step 1: Draw a diagram
Start by drawing a diagram to visualize the given information. Draw a vertical line to represent the tower and label its height as 125 feet. Draw a horizontal line from the base of the tower to represent the ground. Also, draw a line from the top of the tower to the point where the angle is measured (240 feet from the foot of the tower).

Step 2: Identify the right triangle
We can see that we have a right triangle formed by the height of the tower (125 feet), the distance from the foot of the tower to the point where the angle is measured (240 feet), and the side of the hill.

Step 3: Determine the opposite and adjacent sides
In the right triangle, the side of the hill represents the opposite side, and the horizontal distance represents the adjacent side. In this case, the opposite side is the height of the tower (125 feet), and the adjacent side is the horizontal distance (240 feet).

Step 4: Use the tangent ratio
The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, we can use the tangent of the angle to find the angle that the side of the hill makes with the horizontal.

tan(theta) = opposite/adjacent
tan(theta) = 125/240

Step 5: Calculate the angle
Now, use a scientific calculator or a trigonometric table to find the inverse tangent of the above ratio. It will give you the value of the angle.

theta = arctan(125/240)

Using a calculator, the angle comes out to be approximately 28.2 degrees.

Therefore, the angle that the side of the hill makes with the horizontal is approximately 28.2 degrees.