Question

a ladder leans against a vertical wall with the bottom of the ladder 8 ft from the wall on a horizontal floor. at that time the bottom end of the ladder is being pulled away at the rate of 3ft/s and the top of the ladder slips down the wall at a rate of 4 ft/s. how long is the ladder?

Answer 1

Let the height of the ladder be y and the length of the ladder be z:

y^2 + 8^2 = z^2 by Pythagoras

think of it this way. If you pull the ladder 3ft across in one second, then you've pulled the ladder 4ft down in one second, so the height of the new triangle you formed is y - 4 and the width of the new triangle you formed is 11. The length is the same

so (y - 4)^2 + 11^2 = z^2 by Pythagoras

hence y^2 + 8^2 = (y - 4)^2 + 11^2

hence y^2 + 64 = y^2 - 8y + 137

hence 64 = -8y + 137

hence 8y = 63

hence y = 63/8

so the length of the ladder is sqrt(63^2/8^2 + 8^2) = sqrt(63^2 + 8^4)/8

= sqrt(8065)/8

Answer 2

somehow i think my answer's wrong. ah well.