The tops of two poles of height 60 metres and 35 metres are connected by a rope. If the rope makes an angle with the horizontal whose tangent is 5/9 metres, then what is the distance (in metres) between the two poles?

Options

A. 63
B. 30
C. 25
D. 45

Answer

To solve the problem, we need to find the horizontal distance between the two poles given their heights and the tangent of the angle the rope makes with the horizontal.

### Step 1: Understand the Geometry
We have two poles:
- Height of the first pole (h1) = 60 meters
- Height of the second pole (h2) = 35 meters

The difference in height between the two poles is:
\[ h = h1 - h2 = 60 - 35 = 25 \text{ meters} \]

### Step 2: Use the Tangent of the Angle
The problem states that the tangent of the angle (θ) that the rope makes with the horizontal is given as:
\[ \tan(θ) = \frac{5}{9} \]

By definition of tangent in a right triangle:
\[ \tan(θ) = \frac{\text{opposite}}{\text{adjacent}} \]

In our case, the "opposite" side is the difference in height between the two poles (25 meters), and the "adjacent" side is the horizontal distance (d) we want to find.

### Step 3: Set Up the Equation
Using the tangent definition:
\[ \tan(θ) = \frac{h}{d} \]
Substituting the known values:
\[ \frac{5}{9} = \frac{25}{d} \]

### Step 4: Solve for d
Cross-multiplying gives:
\[ 5d = 25 \times 9 \]
\[ 5d = 225 \]
Now, divide both sides by 5:
\[ d = \frac{225}{5} = 45 \text{ meters} \]

### Conclusion
The distance between the two poles is 45 meters.