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?

Draw a diagram.

T = top of tower
B = base of tower
D = point 240' downhill
draw a right triangle with hypotenuse TD. Let E be the right angle.

From the law of sines,

125/sin 25° = 240/sin ∠T
∠T is angle DTB
∠T = 54°
Now you have angle T, since
∠TDB + ∠BDE + ∠DTB = 90°
25° + ∠BDE + 54° = 90°

∠BDE = 11° is the angle you want.

how top draw the figure??

Well, let's start by setting up the situation: we have a tower, a hill, and an angle. Sounds like the beginning of a joke!

Alright, here we go: Why did the side of the hill go to the gym? Because it wanted to get an angle on its fitness!

Now, let's get back to the math behind this angle. To find the angle the side of the hill makes with the horizontal, we can use some trigonometry.

First, let's focus on the right triangle formed by the tower, the hill, and the horizontal. The height of the tower (125 feet) is the opposite side of the angle we're looking for, and the horizontal distance from the foot of the tower to the point where the angle is measured (240 feet) is the adjacent side.

Now, we can use the tangent function to find the angle:

tan(angle) = opposite/adjacent

tan(angle) = 125/240

Now, let's find the actual angle by taking the inverse tangent (arctan) of both sides:

angle = arctan(125/240)

So, with a little bit of calculation, we find that the angle the side of the hill makes with the horizontal is approximately 28.75 degrees.

And there you have it! Not only did you get a joke, but you also got the answer to your question.

To find the angle that the side of the hill makes with the horizontal, we can use trigonometry.

Let's say the angle we are trying to find is θ.

From the given information, we know:
The height of the tower, which is 125 feet,
The distance from the foot of the tower to the point on the hill, which is 240 feet, and
The angle subtended by the tower, which is 25 degrees.

Now, let's draw a right triangle to represent the situation:

|
| /
| /
125 | / θ
| /
| /
|-------------------
240

In this right triangle, the side opposite angle θ is the height of the tower (125 feet), and the side adjacent to angle θ is the distance from the foot of the tower to the point on the hill (240 feet).

Using the tangent function, we have:

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

Now, we can solve for θ by taking the inverse tangent (arctan) of both sides:

θ = arctan(125 / 240)

Using a calculator, we can find that θ is approximately 27.6 degrees.

Therefore, the angle that the side of the hill makes with the horizontal is approximately 27.6 degrees.

To find the angle the side of the hill makes with the horizontal, we can use trigonometric functions.

Let's label the angle we want to find as θ.

First, let's draw a diagram to help solve the problem:

T
/ | \
/ |θ \
/ | \
/ | \
/ | \
/ | \
/ | \
/ |h \
/ | \
/ θ | \
-----------------------------------
240 D B A

In the diagram, T represents the top of the tower, D represents the foot of the tower, and A represents the point on the side of the hill where the tower subtends an angle of 25 degrees.

Given information:
- The height of the tower, TD, is 125 feet.
- The distance from the foot of the tower to point A, AD, is 240 feet.
- The angle θ (which we want to find) is the angle between the side of the hill and the horizontal.

To find the angle θ, we need to use the tangent function:
tan(θ) = opposite/adjacent

In this case:
opposite = TD (the height of the tower) = 125 feet
adjacent = AD (the horizontal distance from the foot of the tower to point A) = 240 feet

Therefore, we can write the equation as:
tan(θ) = 125/240

To find the value of θ, we take the inverse tangent (arctan) of both sides of the equation:
θ = arctan(125/240)

Using a calculator, we can find the approximate value of θ:
θ ≈ 27.92 degrees

Therefore, the angle the side of the hill makes with the horizontal is approximately 27.92 degrees.