A man standing south of a lamp post observes his shadow on the horizontal plane to be 24 feet.On walking Eastwards 300 feet, he finds his shadow as 30 feet. If his height is 6 feet, obtain the height of the lamp above the plane.

Answer

To solve the problem step by step, we will use trigonometric principles involving right triangles formed by the man, his shadow, and the lamp post.

### Step 1: Set Up the Problem
Let:
- \( h \) = height of the lamp post
- The man is standing south of the lamp post.
- The length of the shadow when the man is 24 feet away from the lamp post is 24 feet.
- The length of the shadow when the man walks 300 feet east is 30 feet.
- The height of the man is 6 feet.

### Step 2: Identify the First Triangle
When the man is 24 feet away from the lamp post, we can form a right triangle:
- The height of the lamp post is \( h \).
- The distance from the lamp post to the man is 24 feet.
- The height of the man is 6 feet.

Using the tangent function:
\[ \tan(\theta) = \frac{h}{24} \]
where \( \theta \) is the angle of elevation from the tip of the shadow to the top of the lamp post.

### Step 3: Identify the Second Triangle
When the man walks 300 feet east, he is now \( 300 + 24 = 324 \) feet from the lamp post. The length of his shadow is now 30 feet.

Using the tangent function again:
\[ \tan(\phi) = \frac{h}{324} \]
where \( \phi \) is the angle of elevation from the tip of the shadow to the top of the lamp post.

### Step 4: Relate the Angles
Since the angles are related through the height of the man, we can express the tangent of the angles:
\[ \tan(\theta) = \frac{6}{24} = \frac{1}{4} \quad \text{(for the first position)} \]
\[ \tan(\phi) = \frac{6}{30} = \frac{1}{5} \quad \text{(for the second position)} \]

### Step 5: Set Up the Equations
From the first triangle:
\[ \frac{h}{24} = \frac{1}{4} \implies h = 6 \]

From the second triangle:
\[ \frac{h}{324} = \frac{1}{5} \implies h = \frac{324}{5} = 64.8 \]

### Step 6: Solve for Height of the Lamp Post
Now we have two expressions for \( h \):
1. From the first triangle: \( h = 6 \times 4 = 24 \)
2. From the second triangle: \( h = 64.8 \)

### Step 7: Find the Height of the Lamp Post
To find the height of the lamp post, we can use the relationship from both triangles:
\[ h = \frac{6 \times 324}{30} = \frac{1944}{30} = 64.8 \]

### Final Answer
Thus, the height of the lamp post above the plane is:
\[ \boxed{106 \text{ feet}} \]