equations and derivatives

Question

A street light is at the top of a 18 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 ft from the base of the pole?

Answer 1

the street light is atop a pole 18 feet high

a woman 6 feet tall walks away from the pole and is 30 feet away, walking at 5 feet per second, in a straight line.

How fast is the tip of her shadow moving?

A right triangle is formed with sides 30 + x and 18, where x = the length of her shadow. a similar right triangle is formed with sides 6 and x

x/6 = (30+x)/18 = 30/18 + x/18

x/6 - x/18 = 30/18

2x/18 = 30/18

2x =30

x = 15 feet = the length of her shadow from the woman

the tip of her shadow is 30+15 = 45 feet from the bottom of the pole

this is a "related rates" problem. calculus textbooks have a section on it, usually under a title such as "practical applications of the first derivative" or something similar. the derivative is an instantaneous rate of change. find a formula for the distance of the shadow tip and take the derivative with respect to time.

x +y = the shadow length where y = the woman's distance from the pole. x= y/2, where x is the distance of the tip to the woman

(x+y)' = (y+y/2)' = (3y/2)' = (3/2)y' = (3/2)5 = 7.5 ft/sec = rate of change of the shadow's tip

in one second the woman will be 35 feet from the pole, then

x/6 = (x+35)/18

x = (x+35)/3 = x/3 + 35/3

x-x/3 = 35/3

2x/3 = 35/3

2x =35

x = 17.5

the shadow tip moved 2.5 feet from the woman when the woman moved 5 feet. Or 2.5 +5 = 7.5 feet per second for the tip's speed.

in 2 seconds the woman will be 40 feet from the pole

x/6 = (40+x)/18

x =(40+x)/3 = 40/3 +x/3

2x/3 = 40/3

2x = 40

x = 20

the woman moved ten feet from 30 to 40 feet, as the shadow tip moved 5 feet further fron her, from 15 to 20 feet

the tip moved half her speed, 5/2 = 2.5 + 5 = 7.5 feet per second

one second earlier, she was 25 feet from the pole

x/6 = (25+x)/18

x = (25+x)/3 = 25/3 +x/3

2x/3 = 25/3

2x = 25

x = 12.5

she moved from 25 to 30 feet = 5 feet in one second

while her shadow tip moved from 12.5 feet to 15 feet or 2.5 feet in one second. The tip is moving at 2.5 +5 = 7.5 feet per second

6 seconds earlier she was at the bottom of the pole with zero shadow. in 6 seconds at 5 feet per second she moved 30 feet, while her shadow's tip moved from zero feet to 45 feet in 6 seconds. a tip speed of 45/6 = 7.5 feet per second