Question

A 40 foot tall monument sits on top of a hill. Pete is standing at a point in the hill and observes the top of the monument at an angle elevation of 50 degrees and the bottom of the monument at an angle of elevation of 22 degrees. Find the distance Pete must climb to reach the monument.

Answer 1

The distance Pete must climb to reach the base of the monument is approximately 20.51 feet.

Step-by-step explanation:

To solve this problem, we can use trigonometry, specifically the tangent function, which relates an angle in a right triangle to the ratio of the opposite side (the side across from the angle) to the adjacent side (the side next to the angle). We'll solve this problem in two main steps: first we'll find the distance from Pete to the base of the monument, and second we'll find the height of the hill that Pete must climb.

Let's denote the following:
- `H` as the height of the monument (40 feet).
- `D` as the distance from Pete to the base of the monument (unknown).
- `h` as the height Pete must climb to reach the base of the monument (unknown).
- `α` as the angle of elevation to the bottom of the monument (22 degrees).
- `β` as the angle of elevation to the top of the monument (50 degrees).

First, we can find the distance `D`. We can create a right triangle with `D` as the adjacent side and `h` as the opposite side to angle `α`. Using the tangent function, we get:

tan(α) = h / D
h = D * tan(α)

Next, we use the angle of elevation `β` to the top of the monument to create a right triangle with `D` as the adjacent side and `h + H` as the opposite side. The tangent function gives us:

tan(β) = (h + H) / D
h + H = D * tan(β)

Now we have two equations:
1) h = D * tan(α)
2) h + H = D * tan(β)

We can substitute the value of `h` from the first equation into the second equation:

D * tan(α) + H = D * tan(β)

Now let's solve for `D`:

D * tan(α) + H = D * tan(β)
H = D * tan(β) - D * tan(α)
H = D * (tan(β) - tan(α))

D = H / (tan(β) - tan(α))

Substituting the values we have:

D = 40 / (tan(50°) - tan(22°))

Using a calculator, we can find:

tan(50°) ≈ 1.1918
tan(22°) ≈ 0.4040

Thus:

D = 40 / (1.1918 - 0.4040)
D = 40 / 0.7878
D ≈ 50.77 feet (to two decimal places)

Now that we have `D`, we can find `h` using the equation from the first right triangle with angle `α`:

h = D * tan(α)
h ≈ 50.77 * tan(22°)
h ≈ 50.77 * 0.4040
h ≈ 20.51 feet (to two decimal places)

Thus, the distance Pete must climb to reach the base of the monument is approximately 20.51 feet.

Answer 2

55 feet

Step-by-step explanation:

Please refer to the attached diagram. We assume the distance of interest is the length of segment AC.

The Law of Sines can be used to find the length AC. It is opposite angle B, which is the complement of the elevation angle 50°. The known side of the triangle is AB, which is 40 feet. It is opposite ∠ACB, which is the difference between the elevation angles, an angle of 28°.

The Law of Sines tells us ...

AC/sin(B) = AB/sin(∠ACB)

Multiplying by sin(B), we have ...

AC = AB·sin(B)/sin(∠ACB) = 40·sin(40°)/sin(28°) ≈ 54.767 ft

Pete must climb about 55 feet to reach the base of the monument.

_____

Alternate interpretation

The height of the base of the monument above Pete's observation point is ...

(54.767 ft)·sin(22°) = 20.5 ft

This is the change in elevation required for Pete to reach the monument. The wording "distance Pete must climb" is ambiguous, so we cannot tell if it is the distance Pete must move uphill along the ground, or the change in altitude.