Two vertical poles are on either side of a road. A 30 m long ladder is placed between the two poles. When the ladder rests against one pole, it makes angle `32^(@) 24'` with the pole and when it is turned to rest against another pole, it makes angle `32^(@) 24'` with the road . Calculate the width of the road.

Answer

To solve the problem step by step, we will use trigonometric ratios to find the width of the road between the two poles.

### Step 1: Understand the Setup
We have two vertical poles on either side of a road, and a 30 m long ladder rests against both poles at different angles. The ladder makes an angle of \(32^\circ 24'\) with the first pole and an angle of \(32^\circ 24'\) with the road when resting against the second pole.

### Step 2: Draw the Diagram
Let's denote:
- The first pole as point A.
- The second pole as point C.
- The point where the ladder touches the first pole as point B.
- The point where the ladder touches the ground as point P (for the first pole) and point Q (for the second pole).

### Step 3: Calculate the Height of the Ladder Against the First Pole
Using the triangle formed by the ladder and the first pole (triangle ABP):
- The angle at point B (the angle the ladder makes with the pole) is \(32^\circ 24'\).
- The length of the ladder (hypotenuse) is 30 m.

Using the sine function:
\[ \sin(32^\circ 24') = \frac{PB}{AB} \]
Where \(PB\) is the height of the ladder against the first pole, and \(AB\) is the length of the ladder (30 m).

Rearranging gives:
\[ PB = 30 \cdot \sin(32^\circ 24') \]

Calculating \(PB\):
\[ PB = 30 \cdot 0.536 = 16.08 \text{ m} \]

### Step 4: Calculate the Base Distance from the First Pole
Now, using the cosine function to find the base distance \(BP\):
\[ \cos(32^\circ 24') = \frac{BP}{AB} \]
Rearranging gives:
\[ BP = 30 \cdot \cos(32^\circ 24') \]

Calculating \(BP\):
\[ BP = 30 \cdot 0.844 = 25.32 \text{ m} \]

### Step 5: Calculate the Height of the Ladder Against the Second Pole
For the second pole (triangle CBQ), the angle with the road is also \(32^\circ 24'\). We can use the sine function again:
\[ \sin(32^\circ 24') = \frac{CQ}{BC} \]
Where \(CQ\) is the height of the ladder against the second pole, and \(BC\) is the length of the ladder (30 m).

Rearranging gives:
\[ CQ = 30 \cdot \sin(32^\circ 24') = 30 \cdot 0.536 = 16.08 \text{ m} \]

### Step 6: Calculate the Base Distance from the Second Pole
Using the cosine function again:
\[ \cos(32^\circ 24') = \frac{BQ}{BC} \]
Rearranging gives:
\[ BQ = 30 \cdot \cos(32^\circ 24') = 30 \cdot 0.844 = 25.32 \text{ m} \]

### Step 7: Calculate the Width of the Road
The total width of the road \(PQ\) is the sum of the distances \(BP\) and \(BQ\):
\[ PQ = BP + BQ = 25.32 + 25.32 = 50.64 \text{ m} \]

### Final Answer
The width of the road is \(50.64 \text{ m}\).

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