Street light Calculus Problem

Question

A street light is at the top of a 20-ft-tall pole. A 6-ft tall woman walks away from the pole with a speed of 7 ft/sec along a straight path (see figure). How fast is the tip of her shadow moving, in ft/sec, when she is 35 ft from the base of the pole?

Answer 1

the tip of the shadow moves at 10 feet per second

in the first one second the woman moved 7 feet, while the tip moved 10 feet in that same first second, if she started at the base of the 20 foot pole. the tip moved 10 feet per second.

in 5 seconds, the woman moved 5 sec. x 7 ft/sec. = 35 feet, while the tip moved 50 feet in 5 seconds, a rate of 10 feet per second.

when she's 35 feet from the base of the 20 foot pole

2 right triangles are formed, one with height 20, the other with height 6. They are similar triangles, where the sides have the same ratios, and the angles are all the same.

from the woman to the shadow tip is c feet.

c+35 is the base of the larger triangle with height 20

c is the base of the smaller triangle with height 6

20/(c+35) = 6/c

divide by 2

10/(c+35) = 3/c

cross multiply

3c+105 =10c

7c = 105

divide by 7

c = 105/7 = 15 feet

in 5 seconds the woman moved 35 feet

in 5 seconds the shadow's tip moved 35+15 = 50 feet

50/5 = 10 feet per second = rate of speed of the tip.

when the woman is 35 feet from the pole's base, the tip is moving 10 feet per second. At any distance from the base, the tip moves at 10 feet per second, if she is moving at 7 feet per second.