Question

From a point on the ground, a person notices that a 121 -foot antenna on the top of a hill subtends an angle of 1.5 ∘ . If the angle of elevation to the bottom of the antenna is 25 ∘ , find the height of the hill.

Answer 1

The height of the hill is approximately
`\(171.7\)` feet.

Step-by-step explanation:

Let's denote the height of the hill as
`\(h\)`and the distance from the person to the bottom of the antenna as
`\(d\).`

Using trigonometry, we can set up two equations based on the given information:

1. The tangent of the angle of elevation
`(\(\theta\))` to the bottom of the antenna is given by:

`\[ \tan(\theta) = (h)/(d) \]`

In this case,
`\(\theta = 25^\circ\).`

2. The tangent of the angle subtended by the antenna
`(\(\alpha\))` is given by:

`\[ \tan(\alpha) = \frac{\text{height of antenna}}{\text{distance to antenna}} \]`

In this case,
`\(\alpha = 1.5^\circ\)` and the height of the antenna is
`\(121\)` feet.

Now, solve these equations simultaneously to find
`\(h\):`

`\[ \tan(25^\circ) = (h)/(d) \]`

`\[ \tan(1.5^\circ) = (121)/(d) \]`

Solving the first equation for
`\(d\):`

`\[ d = (h)/(\tan(25^\circ)) \]`

Substitute this expression for
`\(d\)` into the second equation:

`\[ \tan(1.5^\circ) = (121)/((h)/(\tan(25^\circ))) \]`

Now, solve for
`\(h\):`

`\[ h = (121 \cdot \tan(25^\circ))/(\tan(1.5^\circ)) \]`

Calculating this expression gives
`\(h \approx 171.7\)` feet.

Therefore, the height of the hill is approximately
`\(171.7\)` feet.