A street light is at the top of a 15 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 4 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 50 feet from the base of the pole?

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Answer 1 · 2018-05-10T16:37:46

#d'(t_0)=20/3=6,bar6# ft/s

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Using Thales Proportionality theorem for the triangles #AhatOB#, #AhatZH#

The triangles are similar because they have #hatO=90#°, #hatZ=90#° and #BhatAO# in common.

We have #(AZ)/(AO)=(HZ)/(OB)# #<=>#

#ω/(ω+x)=6/15# #<=>#

#15ω=6(ω+x)# #<=>#

#15ω=6ω+6x# #<=>#

#9ω=6x# #<=>#

#3ω=2x# #<=>#

#ω=(2x)/3#

Let #OA=d# then

#d=ω+x=x+(2x)/3=(5x)/3#

For #t=t_0# , #x'(t_0)=4# ft/s

Therefore, #d'(t_0)=(5x'(t_0))/3# #<=>#

#d'(t_0)=20/3=6,bar6# ft/s