Question

a large sphere is on a horizontal field on a sunny day. at a certain time the shadow reaches out a distance of 10 m from the point where the sphere touches the ground. at the same instant a meter stick (held vertically with one end on the ground) casts a shadow of 2 m. what is the radius of the sphere in meters? (assume the sun's rays are parallel and the meter stick is a line segment.)

Answer 1

The radius of the sphere in meters is ,r = $10\sqrt{5} -20$

Think about the angle the ground and the shadow make. Since the sun's beams are parallel, the angle created by the stick's shadow is also equal. Since the stick is 1 m high and its shadow is 2 m long, we know that the stick's angle is arctan 1/2. Therefore, by thinking of a right-angled triangle,

r/10 = tan [arctan(1/2)] = tan (1/2)

Since, tan (θ/2) = 1-cos(θ) / sin(θ)

we find that,

r/10 = $\sqrt{5} -2$

Hence, r = $10\sqrt{5} -20$

So, the radius of the sphere in meters is ,r = $10\sqrt{5} -20$

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