Question

A 30 meter high building sits on top of a hill. The angles of elevation of the top an bottom of the building from the same spot at the base of the hill are measured to be 55 degrees and 50 degrees respectively. How high is the hill to the nearest meter?

I don't understand why it is asking how high the hill is because I though the hill would be the base.

Answer 1

To find the height of the hill, you can use trigonometry. Consider the triangle formed by the hill, the top of the building, and the spot at the base of the hill where the angles of elevation are measured. Use the tangent function to set up equations and solve for the height of the hill.

Step-by-step explanation:

To find the height of the hill, we can use trigonometry. Let's consider the triangle formed by the hill, the top of the building, and the spot at the base of the hill where the angles of elevation are measured. The angle of elevation from the spot to the top of the building is 55 degrees, and the angle of elevation from the spot to the bottom of the building is 50 degrees.

Using the tangent function, we can set up the following equation:
tan(55) = x / h, where x represents the height of the building and h represents the height of the hill.

Similarly, using the tangent function for the angle of elevation to the bottom of the building:
tan(50) = (x + 30) / h.

We can solve these two equations simultaneously to find the height of the hill, h.

Answer 2

A 30 meter high building sits on top of a hill. The angles of elevation of the top an bottom of the building from the same spot at the base of the hill are measured to be 55 degrees and 50 degrees respectively. The height of the hill to the nearest meter is 151 meter.

Step-by-step explanation:

Given, Height of the building on top of a hill = 30 meter

The angel of elevation of the top of the building from a spot at the base of hill = 55 degrees

The angel of elevation of the bottom of the building from the same spot at the base of hill = 50 degrees

The given scenario is shown in the below attachment.

Let us consider the height of hill be 'h' and distance between the base of the hill and the spot of elevation be 'x'.

From the figure,

`tan \, \theta=(h)/(x)\\$\Rightarrow x=(h)/(\tan 50)$\\$\Rightarrow x=(h)/(1.192)$` .... (1)

and
`\tan 55=(30+h)/(x)`

`\Rightarrow x=(30+h)/(\tan 55)`

`\Rightarrow x=(30+h)/(1.192)` .... (2)

[
`\because` tan 50 = 1.192 and tan 55 = 1.428]

Since the distance between the base of the hill and the spot is same for both elevations, equating equations (1) and (2), we get

By cross-multiplying, we get

(30 + h)1.192 = 1.428h

35.76 + 1.192h = 1.428h

1.428h - 1.192h = 35.76

0.236h = 35.76

h = 151.52 meter

Thus, the height of the hill to the nearest meter is 151 meter.