A tower is 120 m high. Its shadow is s m shorter, when the sun's altitude is `60^(@)` than when it was `45^(@)` . Find x correct to nearest metre .

Answer

To solve the problem step by step, we will use trigonometric ratios to find the lengths of the shadows at different angles of elevation of the sun.

### Step 1: Understand the Problem
We have a tower of height 120 m. We need to find the difference in the lengths of the shadows when the angle of elevation of the sun is 45° and 60°. Let the length of the shadow when the angle is 45° be \( BC \) and when the angle is 60° be \( BD \). The difference in the lengths of the shadows is given as \( S = BC - BD \).

### Step 2: Calculate the Shadow Length at 45°
In triangle \( ABC \):
- The height of the tower (perpendicular) = 120 m
- The angle of elevation = 45°

Using the tangent function:
\[ \tan(45°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{120}{BC} \]
Since \( \tan(45°) = 1 \):
\[ 1 = \frac{120}{BC} \implies BC = 120 \text{ m} \]

### Step 3: Calculate the Shadow Length at 60°
In triangle \( ABD \):
- The height of the tower (perpendicular) = 120 m
- The angle of elevation = 60°

Using the tangent function:
\[ \tan(60°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{120}{BD} \]
Since \( \tan(60°) = \sqrt{3} \):
\[ \sqrt{3} = \frac{120}{BD} \implies BD = \frac{120}{\sqrt{3}} = \frac{120 \sqrt{3}}{3} = 40\sqrt{3} \text{ m} \]

### Step 4: Calculate the Difference in Shadow Lengths
Now we find \( S \):
\[ S = BC - BD = 120 - 40\sqrt{3} \]
Substituting the approximate value of \( \sqrt{3} \approx 1.732 \):
\[ S = 120 - 40 \times 1.732 \]
Calculating \( 40 \times 1.732 \):
\[ 40 \times 1.732 = 69.28 \]
Now substituting back:
\[ S = 120 - 69.28 = 50.72 \text{ m} \]

### Step 5: Round the Answer
Rounding \( 50.72 \) to the nearest metre gives us:
\[ S \approx 51 \text{ m} \]

### Final Answer
The value of \( S \) correct to the nearest metre is \( 51 \text{ m} \).

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