The tops of two poles are connected by a wire. The heights of the poles are 10 m and 14 m respectively. If the wire makes a 30° angle with the horizontal, find the length of the wire?

Options

A. 8.5m
B. 8m
C. 7.5m
D. 7 m

Answer

To find the length of the wire connecting the tops of two poles of heights 10 m and 14 m, which makes a 30° angle with the horizontal, we can follow these steps:

### Step 1: Understand the Problem
We have two poles, one of height 10 m and the other of height 14 m. The difference in height between the two poles will help us form a right triangle with the wire as the hypotenuse.

### Step 2: Calculate the Height Difference
The height difference between the two poles is:
\[ \text{Height Difference} = \text{Height of Pole 2} - \text{Height of Pole 1} = 14 \, \text{m} - 10 \, \text{m} = 4 \, \text{m} \]

### Step 3: Set Up the Right Triangle
In the right triangle formed, the height difference (4 m) acts as the opposite side, while the length of the wire (hypotenuse) and the horizontal distance (adjacent side) can be related using trigonometric functions.

### Step 4: Use the Sine Function
Given that the wire makes a 30° angle with the horizontal, we can use the sine function:
\[ \sin(30°) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{4}{L} \]
where \( L \) is the length of the wire.

### Step 5: Solve for the Length of the Wire
We know that:
\[ \sin(30°) = \frac{1}{2} \]
Substituting this into the equation gives:
\[ \frac{1}{2} = \frac{4}{L} \]
Cross-multiplying to solve for \( L \):
\[ L = 4 \times 2 = 8 \, \text{m} \]

### Step 6: Conclusion
The length of the wire connecting the tops of the two poles is:
\[ \boxed{8 \, \text{m}} \]