Understanding the Problem: Lamp Post and Shadow

This question involves a classic geometry problem related to similar triangles. We are given the height of a person, her distance from a lamp post, and the length of her shadow. Our goal is to determine the height of the lamp post.

Let's break down the information provided:

• Person's height (\(H_p\)): 1.5 m
• Distance of the person from the lamp post (\(D\)): 3 m
• Length of the shadow (\(L_s\)): Twice her height.

Calculating Shadow Length

First, we need to calculate the exact length of the person's shadow. The problem states that the length of the shadow is twice her height.

• Person's height \(H_p = 1.5 \text{ m}\)
• Shadow length \(L_s = 2 \times H_p\)
• \(L_s = 2 \times 1.5 \text{ m}\)
• \(L_s = 3 \text{ m}\)

So, the length of the shadow is 3 meters.

Applying Similar Triangles for Lamp Post Height

This problem can be solved by using the concept of similar triangles. Imagine two right-angled triangles formed:

1. The larger triangle: formed by the lamp post, the ground, and the light ray from the top of the lamp to the end of the shadow.
2. The smaller triangle: formed by the person, the ground, and the light ray from the top of the person's head to the end of her shadow.

These two triangles are similar because:

• Both have a right angle with the ground.
• The angle of elevation of the light source (lamp) from the end of the shadow is common to both triangles.

For similar triangles, the ratio of corresponding sides is equal.

Let:

• \(H_l\) = Height of the lamp post (what we need to find)
• \(H_p\) = Height of the person = 1.5 m
• \(D\) = Distance of the person from the lamp post = 3 m
• \(L_s\) = Length of the person's shadow = 3 m

The base of the larger triangle is the distance from the lamp post to the end of the shadow, which is \(D + L_s\).

The base of the smaller triangle is the length of the person's shadow, which is \(L_s\).

Using the proportionality of similar triangles, we can set up the following ratio:

\[ \frac{\text{Height of lamp post}}{\text{Base of large triangle}} = \frac{\text{Height of person}}{\text{Base of small triangle}} \]

\[ \frac{H_l}{D + L_s} = \frac{H_p}{L_s} \]

Step-by-Step Calculation of Lamp Post Height

Now, let's substitute the known values into the equation:

• \(H_l\) = ?
• \(D = 3 \text{ m}\)
• \(L_s = 3 \text{ m}\)
• \(H_p = 1.5 \text{ m}\)

Substitute these values:

\[ \frac{H_l}{3 \text{ m} + 3 \text{ m}} = \frac{1.5 \text{ m}}{3 \text{ m}} \]

Simplify the denominator on the left side and the fraction on the right side:

\[ \frac{H_l}{6 \text{ m}} = \frac{1.5}{3} \]

The fraction \( \frac{1.5}{3} \) simplifies to \( \frac{1}{2} \):

\[ \frac{H_l}{6 \text{ m}} = \frac{1}{2} \]

To find \(H_l\), multiply both sides by 6 m:

\[ H_l = \frac{1}{2} \times 6 \text{ m} \]

\[ H_l = 3 \text{ m} \]

Therefore, the height of the lamp post is 3 meters.

Final Answer Summary

By applying the principle of similar triangles and carefully substituting the given measurements, we found that the height of the lamp post is 3 meters.