how fast is the top of the ladder moving down the wall

Question

A 6 foot ladder, resting against a wall, begins to slip down the wall. When the angle of the ladder with the ground is 35 degrees, the bottom of the ladder is moving away from the wall at 0.5 ft/s. At that moment, how fast is the top of ladder moving down the wall?

Answer 1

the ladder, wall and ground form a right triangle

with hypotenuse = h = 6

altitude = a = 6sin35

base = b = 6cos35

by the Pythagorean Theorem hypotenuse squared = sum of squares of the other 2 sides

h^2 = a^2 + b^2

take the derivative with respect to time

2hh' =2aa' + 2bb'

divide by 2

hh' = aa' +bb'

solve for a'

a' = (hh'-bb')/a

h'=0, so

a' = (0-bb')/a

a' = -bb'/a, plug in values for b,b' and a

a' = -6cos35(.5)/6sin35

a' = -.5cot35

a'=about - 0.714074 feet per second

or = about -0.7 ft/s

Answer 2

x² + y² = 36; 2xdx/dt + 2ydy/dt = 0; y/x = tan35°; y = xtan35°; x²sec²35° = 36; x = 6cos35°; y = 6sin35°;

dy/dt = - x/y·dx/dt = - dx/dt·cot35° = - 0.5/tan35° = - 0.77141 ft/s