As shown in the given figure, 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.
Show Hint
Always ensure all units are consistent (convert cm to m) before substituting values into similarity ratios.
Step 1: Understanding the Concept:
This problem involves similar triangles. The lamp post, the girl, and their shadows form triangles with the same angle of elevation from the ground. Step 2: Detailed Explanation:
Let AB be the lamp post and CD be the girl. Let AE be the ground level and \( x \) be the shadow length DE.
Height of lamp post (\( AB \)) = 3.6 m.
Height of girl (\( CD \)) = 90 cm = 0.9 m.
Speed of girl = 1.2 m/s. Time = 4 s.
Distance walked by girl (\( BD \)) = \( \text{Speed} \times \text{Time} = 1.2 \times 4 = 4.8 \text{ m} \).
In \( \triangle ABE \) and \( \triangle CDE \):
\( \angle B = \angle D = 90^\circ \)
\( \angle E = \angle E \) (Common)
So, \( \triangle ABE \sim \triangle CDE \) by AA similarity.
The ratio of corresponding sides:
\[ \frac{AB}{CD} = \frac{BE}{DE} \]
\[ \frac{3.6}{0.9} = \frac{4.8 + x}{x} \]
\[ 4 = \frac{4.8 + x}{x} \]
\[ 4x = 4.8 + x \]
\[ 3x = 4.8 \implies x = 1.6 \text{ m} \]. Step 3: Final Answer:
The length of her shadow after 4 seconds is 1.6 m.
