Theory question 105
TrigonometryA boy 1.2 m tall, stands 6 m away from the foot of a vertical lamp pole 4.2 m long. If the lamp is at the tip of the pole, (a) represent this information in a diagram. (b) calculate the: (i) length of the shadow of the boy cast by the lamp; (ii) angle of elevation of the lamp from the boy, correct to the nearest degree.
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(a) Sketch: pole 4.2 m and boy 1.2 m standing 6 m apart on level ground; a line from the lamp over the boy's head to the ground marks the end of the shadow s.
(b)(i) Similar triangles: 1.2/s = 4.2/(6 + s) → 7.2 + 1.2s = 4.2s → s = 2.4 m.
(ii) From the top of the boy's head: tan θ = (4.2 − 1.2)/6 = 0.5 → θ ≈ 27°.
(Measured from the boy's feet: tan θ = 4.2/6 → θ ≈ 35°.)
The lamp, the top of the boy's head and the tip of the shadow lie on one straight line.