The upper `(3/4)^(th)` portion of a vertical pole subtends an angle `"tan"^(-1)3/5` at a point in the horizontal plane through its foot at a distance 40 m from the foot . A possible height of the vertical pole is

Options

A. 80 m
B. 20 m
C. 40 m
D. 60 m

Answer

To solve the problem step-by-step, we will use trigonometric relationships and the information provided in the question.

### Step 1: Understand the Problem
We have a vertical pole of height \( h \). The upper \( \frac{3}{4} \) portion of the pole subtends an angle of \( \tan^{-1} \left( \frac{3}{5} \right) \) at a point 40 m away from the foot of the pole. We need to find the height \( h \) of the pole.

### Step 2: Define the Variables
Let:
- \( AB \) be the height of the pole \( h \).
- \( CD \) be the upper \( \frac{3}{4} \) portion of the pole, which is \( \frac{3}{4}h \).
- \( AD \) be the lower \( \frac{1}{4} \) portion of the pole, which is \( \frac{1}{4}h \).
- The distance from the foot of the pole to the point on the ground is \( 40 \) m.

### Step 3: Set Up the Trigonometric Relationship
The angle \( D \) subtended by the upper \( \frac{3}{4} \) portion at point \( C \) can be expressed in terms of tangent:
\[ \tan D = \frac{CD}{AC} = \frac{\frac{3}{4}h}{40} \]
We know from the problem that \( D = \tan^{-1} \left( \frac{3}{5} \right) \), so:
\[ \tan D = \frac{3}{5} \]

### Step 4: Equate the Two Expressions for Tangent
Now we can set the two expressions for \( \tan D \) equal to each other:
\[ \frac{\frac{3}{4}h}{40} = \frac{3}{5} \]

### Step 5: Solve for \( h \)
Cross-multiplying gives:
\[ 3 \cdot 40 = \frac{3}{4}h \cdot 5 \]
\[ 120 = \frac{15}{4}h \]
Multiplying both sides by \( \frac{4}{15} \):
\[ h = \frac{120 \cdot 4}{15} = \frac{480}{15} = 32 \text{ m} \]

### Step 6: Verify the Calculation
To ensure the calculation is correct, we can double-check:
1. \( \frac{3}{4}h = \frac{3}{4} \cdot 32 = 24 \) m.
2. \( \tan D = \frac{24}{40} = \frac{3}{5} \), which matches the given angle.

### Conclusion
The possible height of the vertical pole is:
\[ \boxed{32 \text{ m}} \]