Question

A 13-ft ladder leans against a

wall. The bottom of the ladder
is 5 ft from the wall. The bottom
is then pulled out 4 ft farther.
How much does the top end
move down the wall?

Answer 1

(12 - 2 √22) ft

Explanation:

by Pythagoras' Theorem:

in right-angled triangle, the square of the hypotenuse will equal the sum of the squares of the other two sides.

call the ladder L, call the distance of bottom of ladder from bottom of wall G, call the vertical height of the wall where the top of ladder meets it H.

we have L² = G² + H²

H² = L² - G²

= 13² - 5²

= 169 - 25

= 144

H = √144 = 12.

the ladder is opposite the right-angle, ie it's the hypotenuse.

the ladder is 5ft from the wall.

if bottom of ladder is pulled out 4ft more, this reduces the height H.

the length of ladder L remains the same (can't compress a ladder)

G, the floor distance, is now 5 + 4 = 9ft

H² = L² - G²

= 169 - 9²

= 169 - 81

= 88

H = √88

= √(4 X 22)

= √4 X √22

= 2√22

The vertical height, H, was 12 ft. it's now 2 √22

so it has moved down the wall (12 - 2 √22) ft.