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.
Step-by-Step Solution
Key Concept: The lamp, the top of the girl and the tip of the shadow form two similar right‑angled triangles. By using the property of similar triangles, the ratio of corresponding sides gives a relation between the distance of the girl from the lamp and the length of her shadow.
1. Convert all quantities to the same unit\
Height of girl = 90 cm = 0.9 m.\
Height of lamp = 3.6 m (already in metres).\
2. Find the distance of the girl from the lamp after 4 s\
Speed = 1.2 m/s, time = 4 s\
\[ d = vt = 1.2 \times 4 = 4.8\ \text{m} \]\
So the girl is 4.8 m away from the base of the lamp.
3. Set up the similar‑triangle relation\
Let \(x\) be the length of the shadow (in metres).\
\[\frac{\text{Height of lamp}}{\text{Distance from lamp to tip of shadow}} = \frac{\text{Height of girl}}{\text{Length of shadow}}\]\
The distance from the lamp to the tip of the shadow = \(d + x = 4.8 + x\).\
Hence,\
\[ \frac{3.6}{4.8 + x} = \frac{0.9}{x} \]
4. Solve for \(x\)\
Cross‑multiply: \(3.6x = 0.9(4.8 + x)\).\
Expand: \(3.6x = 4.32 + 0.9x\).\
Bring like terms together: \(3.6x - 0.9x = 4.32\) → \(2.7x = 4.32\).\
\[ x = \frac{4.32}{2.7} = 1.6 \ \text{m} \]
5. State the answer\
The length of the girl's shadow after 4 seconds is 1.6 metres.
Correct Answer: 1.6 m