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?

Question

2010 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-10T16:37:46

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

enter image source here

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