Since we are looking for the rate of change of the angle, we need
an equation that contains a trig relationship with the angle.
Let the angle at the base of the ladder be θ.
If the height up the wall is y and the distance of the base of the ladder
to the wall is x, then
x^2 + y^2 = 25^2
for the given case, x = 15
15^2 + y^2 = 25^2
y = 20
given: dy/dt = -3 ft/s when x = 15 and y = 20
find dθ/dt at that instant.
I choose
sinθ = y/25 , since we need dθ/dt and we want to involve the given dy/dt
25 sinθ = y
25cosθ dθ/dt = dy/dt
dθ/dt = -3/(25cosθ) = -3/(15/25) = -5
the angle is decreasing at 5 radians/sec
(the rate seems a bit big, better check my arithmetic
A ladder 25 feet long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall so that the top of the ladder is sliding down at 3ft/sec, how fast is
the measure of the measure of the acute angle between the ladder and the ground changing when
the bottom of the ladder is 15 feet from the wall?
1 answer
2 answers