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Extension Question - Arcs / Radians (1 Viewer)

ItsNotHSC

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Anyone know how to solve this one (q20 11I Cambridge Extension 1 Year 11):

A certain hill is represented by a hemisphere of a radius 1km. A man 180cm tall walks down the hill from the summit S at 6km/h. How long (correct to the nearest second) will it be before he is invisible to a person lying on the ground at S?

diagrahm.PNG
 

CM_Tutor

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Consider the given diagram:

View attachment 31809

Let be the centre of the arc / hill.

Let the right angle shown in the diagram be at and let be the point on the hill that is 1.8 m vertically below .

Let produced meet the "base" of the arc at .

Let the arc subtend an angle at , so that .

Now, we know that and that (as is a rectangle), and so .

Further, (alternate angles on parallel lines, )

So,

Thus, the length of the arc

Walking at , the time taken to travel distance is
 

Lith_30

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let arc length
let that hypotenuse created by the right angle triangle
let the angle subtended the origin from point S and the final position of the man
using radius is 1km so
using the cosine rule


also using the right angle triangle created in the diagram




Using the two equations created we sub them in together



square both sides

using double angle formula

trig identities



divide both sides by 4 and multiply both sides by sin(θ/2)

power both sides by 1/4





We know that

which is the distance that the man walked

Using time = distance/speed

convert into seconds



seconds
 

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