How long is the ladder?

Question

The top of a ladder slides down a vertical wall at a rate of 0.375 m/s. At the moment when the bottom of the ladder is 6 m from the wall, it slides away from the wall at a rate of 0.5 m/s. How long is the ladder?

Answer 1

a' = -.375 = 3/8 m/sec

b'= .5 = 1/2 m/sec

b=6 m

L'=0

use the Pythagorean Theorem

L^2 = a^2 + b^2

2LL' = 2aa' + 2bb'

divide by 2 and substitute the values for L'=0, a'=-.375, b'= .5

0 = 3a/8 + 6(.1/2)

3a/8 = 3

a= 8 meters off the ground for tip of the Ladder

L^2 = 8^2 + 6^2

= 64+36 =100

L =sqr100

L = 10 meters Long Ladder

it's a 3-4-5 right triangle

multiplied by factor of 2

6-8-10 meters

angle opposite 10 is 90 degrees. sin90 = 1

angle opposite 8 is B, the angle of the ladder with the ground

B can be found by the of Law of Sines

L/sin90 = 8/sinB

10/1 = 8/sinB

sinB = 8/10 = 4/5= .8

B = sin^-1(.8) = about 23.58 degrees

L= 8/sinB = 8/.8= 10