Related Rates Problem

Question

A 18 ft ladder leans against a wall. The bottom of the ladder is 4 ft from the wall at time 𝑡=0 and slides away from the wall at a rate of 3 ft/sec. 



Find the velocity of the top of the ladder at time 𝑡=2.

The velocity of ladder at time 𝑡=2 is ___________  ft/sec.

Answer 1

x² + y² = 18²

x² + y² = 324

Take the derivative of both sides with respect to t:

2x(dx/dt) + 2y(dy/dt) = 0

2y(dy/dt) = -2x(dx/dt)

dy/dt = -2x/2y(dx/dt)

dy/dt = -x/y(dx/dt)

For t = 2, calculate x:

x(t) = (dx/dt)(t) + x_i so x(2) = (3)(2) + 4 = 10 ft

If x = 10, then y = √(324 - 10²) = √224 = 4√14

Using the previous equation dy/dt = -x/y(dx/dt),

dy/dt = -(10*3)/(4√14) = -15/(2√14) = -(15√14)/28 = (approx) 2 ft/s