A man 170cm tall walks away from a street light 3 m high at the rate of 2m/s. Find rate shadow is growing?

Answer 1

Placing the man an arbitrary distance from the light, say 1 m, the angle between the top of his shadow and the top of the light is equal to the angle created in the triangle between the top of the mans head and the top of the light, the tan of which is is 1.3 / 1 = 1.3. Therefore 1.3 is = 3 / x, where x is the distance between the top of the mans shadow and the bottom of the light which is approx 2.3 m. Therefore the length of the shadow = 1.3 m, 2.3 - 1 (the distance between the man and the light).

Repeating this process for a distance of 3 m away, an extra second of walking if the man is walking 2 m/s, gives a shadow distance of approx 3.92 m. Therefore the shadow increases by 2.62 m every second.

Answer: 2.62 m /s

Answer 2

At any distance from the light we have two similar triangles. The larger one consists of a vertical from the light, the horizontal distance from the vertical to the end of the shadow, and the distance from the end of the shadow to the light. The other consists of man's height, his shadow length, and dist. from the end of the shadow to the man's head from the vertical to the end of the man's shadow, and the distance from the light to the end of the shadow. The altitude of the smaller triangle is 1.7 m. The altitude of the larger triangle is 3 m. The ratio of the shadow length to the horizontal distance from the light to the end of the shadow is always 1.7/3. The ratio of the shadow length to the man's horizontal distance from the light is 1.7/1.3. Thus the shadow grows at 2*1.7/1.3 = 2.615 m/s.