A spotlight at a building corridor is fastened to a wall 8m above the floor. A lady 1.75m tall moves away from the wall at a speed of 0.75m/s. a.)at what rate is the lenght of her shadow increasing? b.)at what speed is the tip of her shadow moving?

coexpennan 2021-02-27 Answered
A spotlight at a building corridor is fastened to a wall 8m above the floor. A lady 1.75m tall moves away from the wall at a speed of 0.75ms.
a.)at what rate is the lenght of her shadow increasing?
b.)at what speed is the tip of her shadow moving?
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Expert Answer

Theodore Schwartz
Answered 2021-02-28 Author has 99 answers

Step 1
Please have a look at the picture below to understand what's going on:
image
For the sake of clarity,
s = length of the shadow = AD
and x = distance of the lady from the wall = BD
dxdt=0.75ms
Step 2
Triangle ADE is similar to triangle ABC (AAA criterion of similarity)
Hence, DEBC=ADAB (Corresponding sides of similar triangles are proportional)
Hence, 1.758=ss+x
Hence, 1.75(s+x)=8s
Or, 1.75x=(81.75)s=6.25s
Step 3
Part (a)
Differentiate both sides w.r.t time t to get:
1.75dxdt=6.25dsdt
Hence, the rate at which length of her shadow is increasing = dsdt=(1.756.25)dxdt=(1.756.25)x0.75=0.21ms
Step 4
Part (b)
the rate at which the tip of her shadow is moving = rate t which she is moving + rate at which the length of the shadow is increasing = dxdt+dsdt=0.75+0.21=0.96ms
Step 5
Final answers:
Part (a) 0.21ms
Part (b) 0.96ms

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