The tops of two poles of height 40 m and 25 m are connected by a wire. If the wire makes an angle `30^(@)` with the horizontal, then the length of the wire is

Options

A. 30 m
B. 20 m
C. 15 m
D. 25 m

Answer

To find the length of the wire connecting the tops of two poles of heights 40 m and 25 m, we can follow these steps:

### Step 1: Determine the vertical distance between the tops of the poles.
The height of the first pole is 40 m and the height of the second pole is 25 m. The vertical distance (h) between the tops of the two poles is:
\[ h = 40 \, \text{m} - 25 \, \text{m} = 15 \, \text{m} \]

**Hint:** Subtract the height of the shorter pole from the height of the taller pole to find the vertical distance.

### Step 2: Use the angle of the wire with the horizontal.
The wire makes an angle of \(30^\circ\) with the horizontal. We can visualize this as a right triangle where:
- The vertical side is the height difference (15 m).
- The hypotenuse is the length of the wire (L).
- The angle with the horizontal is \(30^\circ\).

### Step 3: Apply the sine function.
In a right triangle, the sine of an angle is defined as the ratio of the opposite side to the hypotenuse. Therefore, we can write:
\[ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L} \]
Substituting the known values:
\[ \sin(30^\circ) = \frac{15}{L} \]

### Step 4: Substitute the value of \(\sin(30^\circ)\).
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
So, we can substitute this into our equation:
\[ \frac{1}{2} = \frac{15}{L} \]

### Step 5: Solve for L.
Cross-multiplying gives:
\[ 1 \cdot L = 2 \cdot 15 \]
\[ L = 30 \, \text{m} \]

### Conclusion:
The length of the wire connecting the tops of the two poles is:
\[ \boxed{30 \, \text{m}} \]