Rate of shadow !!??
Calculus
Level
2
A man of height 2 m walks at a uniform speed of 5 km/hr away from a lamp post which is 6 m high. Find the rate at which the length of his shadow increases.
The answer is 2.5.
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1 solution
Since the man and its shadow will create a triangle proportional to the triangle created by the lamp post and the sum of the shadow plus its distance from the lamp post, then: Let x = distance of man/shadow from the lamp post y = length of shadow dx/dt = rate of increase in the distance of man from lamp post dy/dt = rate of increase in shadow's length
We are already given dx/dt to be 5 kph.
Using ratio and proportion, we can get the equation: 6/(x+y) = 2/y
Simplifying: y = x/2
Differentiating the whole equation by time: dy/dt = (dx/dt)/2
dy/dt = 5 kph / 2
dy/dt = 2.5 kph
Thus, the rate of increase in shadow's length is 2.5 kph.