A surveyor wants to find the height of the top of a hill. He observes that the angles of elevation of the top of the hill at points C and D, 300m apart, lying on the base of the hill and on the same side of the hill are 30° and 45° respectively What is the height of the hill.

Question

A surveyor wants to find the height of the top of a hill. He observes that the angles of elevation of the top of the hill at points C and D, 300m apart, lying on the base of the hill and on the same side of the hill are 30° and 45° respectively What is the height of the hill.

Answer 1

Let AB be the height of the mountain, where A is at the top and B is on the line containing C and D

let BD = x, and because of the 45° angle in the right-triangle ADB, AB = x

then by Pythagoras AD = √2 x

In triangle ACD , angle C 30, angle CDA = 135, leaving 15° for angle CAD

by the sine law
√2 x/sin30 = 300/sin15
x = 300 sin30/(√2 sin15)
= ...

you do the button pushing , (I got appr 409.8)

Answer 2

To find the height of the hill, we can use the concept of trigonometry and the given angles of elevation.

Let's assume that the height of the hill is represented by 'h'.

From the information given, we have two right-angled triangles: ΔABC and ΔABD.

In triangle ΔABC, angle BAC is 90° and angle ABC is 30°.
In triangle ΔABD, angle BAD is 90° and angle ABD is 45°.

Let's analyze the triangle ΔABC first:

Step 1: Determine the length of side BC.
Since triangle ΔABC is a right-angled triangle, we can use trigonometry to find the length of side BC.
We know that the tangent of angle ABC (30°) is equal to the ratio of the opposite side (h) to the adjacent side (BC).
So, tan(30°) = h / BC.

Step 2: Calculate the length of side BC.
Rearrange the equation to isolate BC:
BC = h / tan(30°).

Now, let's analyze the triangle ΔABD:

Step 3: Determine the length of side BD.
Again, using trigonometry, we can find the length of side BD.
The tangent of angle ABD (45°) is equal to the ratio of the opposite side (h) to the adjacent side (BD).
Therefore, tan(45°) = h / BD.

Step 4: Calculate the length of side BD.
Rearrange the equation to isolate BD:
BD = h / tan(45°).

Given that points C and D are 300m apart:

Step 5: Find h, the height of the hill.
Since BC and BD are the distances from the base of the hill to points C and D, respectively, we have:
BC + BD = 300.
Substituting the values we found in steps 2 and 4, we get:
(h / tan(30°)) + (h / tan(45°)) = 300.

Solving this equation will give us the height of the hill (h).

Answer 3

To find the height of the hill, we can use the concept of trigonometry and the information given.

Let's consider the triangle formed by the top of the hill (point A), point C, and point D.

First, let's draw a diagram to better visualize the problem.

A
/|
/ |
c / |h
/ |
/ |
C-----D

In this triangle, angle CAD is 45°, angle CAB is 30°, and CD is given as 300m.

Now, let's consider the tangent function.

The tangent of an angle is equal to the opposite side divided by the adjacent side.

In our case, we want to find the height of the hill (h), which is the opposite side to the angle CAB, and the adjacent side is CD.

So, we have the equation:

tan(30°) = h / 300m

To find h, we can rearrange the equation:

h = tan(30°) * 300m

Now, let's calculate the value of h.

Using a scientific calculator, find the tangent of 30°:

tan(30°) ≈ 0.5774

Now, substitute this value back into the equation:

h = 0.5774 * 300m

Calculating further:

h ≈ 173.22m

Therefore, the height of the hill is approximately 173.22 meters.