Calculus Question

Question

A spotlight on the ground shines on a wall 12m away. If a man 2m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4m from the building?

Answer 1

spotlight shines on a wall 12 meters away

man 4 meters from the wall walks towards the wall at a speed of 1.6 m/sec

man is 2 meters tall

how fast is his shadow on the wall decreasing??

construct 2 right triangles, one inside and congruent to the larger

for the larger

base = b = 12 m

b' = 1.6

length of man's shadow on the wall = a

light beam's length from man's head to the wall = h= 3sqr17

find a' = speed of shadow on wall

smaller has base=12-4=8 m

height=2 m

hypotenuse= sqr68=2sqr27

h'= about -1.6745 m/s

it's a related rates problem. plug in known values and solve for a'

a'=(hh'-bb')/a= h'sqr17- 6.4

in one second a goes from 3 to 2.5 m, a'= -1/2 m/sec