Question

3. A person standing 3 feet away from the base of a lamp post casts a shadow 4 feet long. If the person is 6 feet tall and walks away from the lamp post at a speed of 400 feet per minute, at what rate will his shadow lengthen

Answer 1

533.33 ft/min

Explanation:

We are given that

x=3 feet

y=4 feet

Height of person,h=6 feet

`(dx)/(dt)=400 ft/min`

Triangle ABC and A'B'C are similar because all right triangles are similar

By using similarity property

`(h')/(6)=(7)/(4)`

`h'=10.5 feet`

`x+y=3+4=7 feet`

`(h')/(6)=(x+y)/(y)`

`(10.5)/(6)=(x+y)/(y)`

`10.5y=6x+6y`

`10.5y-6y=6x`

`4.5y=6x`

Differentiate w.r.t t

`4.5(dy)/(dt)=6(dx)/(dt)`

`(dy)/(dt)=(6* 400)/(4.5)ft/min`

`(dy)/(dt)=533.33ft/min`

Hence, If the person is 6 feet tall and walks away from the lamp post at a speed of 400 feet per minute then his shadow moving at the rate 533.33 ft/min