A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. [Figure: lamp-post AB vertical, girl CD vertical standing away from the post; ray from lamp top A grazes girl's head C and meets ground at E; DE is the shadow.]

Step 1: Convert girl's height to meters

The girl's height is given in centimeters, but all other lengths are in meters. To maintain consistency, we convert the girl's height to meters. Since $1 \text{ m} = 100 \text{ cm}$, we divide $90 \text{ cm}$ by $100$ to get $0.9 \text{ m}$.

Step 2: Calculate distance walked by the girl

The girl is walking away from the lamp-post at a constant speed. We need to find the distance she has covered in the given time. We use the formula: distance equals speed multiplied by time. This distance is represented as BD in the diagram.

Step 3: Calculate the value of BD

Multiplying the speed $1.2 \text{ m/s}$ by the time $4 \text{ s}$, we find that the distance travelled by the girl from the base of the lamp-post is $4.8 \text{ meters}$.

Step 4: Identify similar triangles

We have two right-angled triangles: $\triangle ABE$ (formed by the lamp-post, the ground, and the light ray) and $\triangle CDE$ (formed by the girl, the ground, and her shadow). Both the lamp-post and the girl are perpendicular to the ground, so $\angle B = \angle D = 90^\circ$. Also, $\angle E$ is common to both triangles. Therefore, by the AA similarity criterion, $\triangle ABE$ is similar to $\triangle CDE$.

Step 5: Set up proportionality of sides

Since $\triangle ABE \sim \triangle CDE$, the ratio of their corresponding sides must be equal. Here, $AB$ is the height of the lamp-post, $CD$ is the height of the girl, $BE$ is the total distance from the lamp-post to the end of the shadow, and $DE$ is the length of the shadow.

Step 6: Substitute known values

We substitute the given values into the proportion. The height of the lamp $AB = 3.6 \text{ m}$, the height of the girl $CD = 0.9 \text{ m}$. The distance $BE$ is the sum of $BD$ (distance girl walked) and $DE$ (length of shadow), so $BE = 4.8 + DE$. Let $DE$ be $x$ for calculation ease.

Step 7: Simplify and solve for DE

Simplify the ratio $\frac{3.6}{0.9}$ to $4$. Then, multiply both sides by $DE$ to eliminate the denominator. This yields a linear equation $4 \times DE = 4.8 + DE$. Subtract $DE$ from both sides ($4 \times DE - DE = 3 \times DE = 4.8$). Finally, divide $4.8$ by $3$ to solve for $DE$.

Step 8: Calculate the length of the shadow

Performing the division, $4.8 \div 3$ equals $1.6$. Thus, the length of the girl's shadow after 4 seconds is $1.6$ meters.

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