Shadow Length B4D9B2

1. **State the problem:** A girl of height $90$ cm (which is $0.9$ m) walks away from a lamp-post that is $3.6$ m tall at a speed of $1.2$ m/s. We want to find the length of her shadow after $4$ seconds. 2. **Identify variables and formula:** Let $x$ be the distance of the girl from the lamp-post after $t$ seconds. Let $s$ be the length of the shadow. Since the girl walks at $1.2$ m/s, after $4$ seconds: $$x = 1.2 \times 4 = 4.8 \text{ m}$$ 3. **Use similar triangles:** The lamp-post, the girl, and their shadows form two similar right triangles: - Large triangle: height $3.6$ m, base $x + s$ - Small triangle: height $0.9$ m, base $s$ By similarity: $$\frac{3.6}{x + s} = \frac{0.9}{s}$$ 4. **Solve for $s$:** Cross-multiply: $$3.6 s = 0.9 (x + s)$$ $$3.6 s = 0.9 x + 0.9 s$$ Bring terms involving $s$ to one side: $$3.6 s - 0.9 s = 0.9 x$$ $$2.7 s = 0.9 x$$ Divide both sides by $2.7$: $$s = \frac{0.9 x}{2.7}$$ Simplify fraction: $$s = \frac{\cancel{0.9} \times x}{\cancel{2.7} \times 3} = \frac{x}{3}$$ 5. **Calculate $s$ using $x=4.8$ m:** $$s = \frac{4.8}{3} = 1.6 \text{ m}$$ **Final answer:** The length of the girl's shadow after $4$ seconds is $1.6$ meters.