If the shadow of a tower is 30 m when the sun's altitude is `30^(@)` what is the length of the shadow when the sun's altiutude is `60^(@)`?

Options

A. `10sqrt2m`
B. 20 m
C. 10 m
D. 12 m

Answer

To solve the problem step by step, we will use trigonometric ratios.

### Step 1: Understand the scenario
We have a tower (let's call it point A) and the shadow of the tower on the ground (let's call the end of the shadow point C). The length of the shadow (BC) is given as 30 meters when the sun's altitude is 30 degrees.

### Step 2: Set up the triangle for the first scenario
In triangle ABC:
- Angle ACB (the angle of elevation) = 30 degrees
- BC (the length of the shadow) = 30 m
- AB (the height of the tower) = ?

Using the tangent function:
\[ \tan(30^\circ) = \frac{AB}{BC} \]

### Step 3: Substitute the known values
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
So we can write:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{30} \]

### Step 4: Solve for AB (height of the tower)
Cross-multiplying gives:
\[ AB = 30 \cdot \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} \text{ meters} \]

### Step 5: Set up the triangle for the second scenario
Now, we need to find the length of the shadow (BD) when the sun's altitude is 60 degrees. In triangle ABD:
- Angle ADB (the angle of elevation) = 60 degrees
- AB (the height of the tower) = \(\frac{30}{\sqrt{3}}\)
- BD (the new length of the shadow) = ?

Using the tangent function again:
\[ \tan(60^\circ) = \frac{AB}{BD} \]

### Step 6: Substitute the known values
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
So we can write:
\[ \sqrt{3} = \frac{\frac{30}{\sqrt{3}}}{BD} \]

### Step 7: Solve for BD (length of the shadow)
Cross-multiplying gives:
\[ BD \cdot \sqrt{3} = \frac{30}{\sqrt{3}} \]
Now, solving for BD:
\[ BD = \frac{30}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} = \frac{30}{3} = 10 \text{ meters} \]

### Final Answer
The length of the shadow when the sun's altitude is 60 degrees is **10 meters**.

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