At a particular time , when the sun's altitude is `30^(@)` , the length of the shadow of a vertical tower is 45 m . Calculate :
(i) the height of the tower ,
(ii) the length of the shadow of the same tower, when the sun's altitude is :
(a) `45^(@)` , (b) `60^(@)`

Answer

To solve the problem step by step, we will use trigonometric ratios, specifically the tangent function, which relates the height of the tower to the length of its shadow based on the angle of elevation of the sun.

### Step 1: Calculate the height of the tower when the sun's altitude is 30 degrees.

Given:
- Angle of elevation (θ) = 30 degrees
- Length of the shadow (L) = 45 m

Using the tangent function:
\[ \tan(θ) = \frac{\text{Height of the tower (h)}}{\text{Length of the shadow (L)}} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{h}{45} \]

From trigonometric tables, we know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

So we can write:
\[ \frac{1}{\sqrt{3}} = \frac{h}{45} \]

Now, cross-multiplying gives:
\[ h = 45 \times \frac{1}{\sqrt{3}} = \frac{45}{\sqrt{3}} = \frac{45 \sqrt{3}}{3} = 15\sqrt{3} \]

Calculating the numerical value:
\[ h \approx 15 \times 1.732 = 25.98 \text{ m} \]

### Step 2: Calculate the length of the shadow when the sun's altitude is 45 degrees.

Now, we need to find the length of the shadow (x) when the angle of elevation is 45 degrees.

Given:
- Angle of elevation (θ) = 45 degrees
- Height of the tower (h) = 25.98 m

Using the tangent function again:
\[ \tan(45^\circ) = \frac{h}{x} \]

Since:
\[ \tan(45^\circ) = 1 \]

We can write:
\[ 1 = \frac{25.98}{x} \]

Cross-multiplying gives:
\[ x = 25.98 \text{ m} \]

### Step 3: Calculate the length of the shadow when the sun's altitude is 60 degrees.

Now, we need to find the length of the shadow (x) when the angle of elevation is 60 degrees.

Given:
- Angle of elevation (θ) = 60 degrees
- Height of the tower (h) = 25.98 m

Using the tangent function:
\[ \tan(60^\circ) = \frac{h}{x} \]

Since:
\[ \tan(60^\circ) = \sqrt{3} \]

We can write:
\[ \sqrt{3} = \frac{25.98}{x} \]

Cross-multiplying gives:
\[ x = \frac{25.98}{\sqrt{3}} \]

Calculating the numerical value:
\[ x \approx \frac{25.98}{1.732} \approx 15 \text{ m} \]

### Final Answers:
(i) The height of the tower is approximately **25.98 m**.
(ii) The lengths of the shadow are:
(a) **25.98 m** when the sun's altitude is 45 degrees.
(b) **15 m** when the sun's altitude is 60 degrees.