Answer: the top of the ladder is sliding down the bricks at - 1.25 ft/s
Step-by-step explanation:
The ladder forms a right angle triangle with the ground. The length of the ladder represents the hypotenuse.
Let x represent the distance from the top of the ladder to the ground(opposite side)
Let y represent the distance from the foot of the ladder to the base of the wall(adjacent side)
The bottom of the ladder is pulled away at the rate of 3ft/sec. This means that y is increasing at the rate of 3ft/sec. Therefore,
dy/dt = 3 ft/s
The rate at which x is reducing would be
dx/dt
Applying Pythagoras theorem which is expressed as
Hypotenuse² = opposite side² + adjacent side², it becomes
x² + y² = 13²- - - - - - - -1
Differentiating with respect to time, it becomes
2xdx/dt + 2ydy/dt = 0
2xdx/dt = - 2ydy/dt
Dividing through by 2x, it becomes
dx/dt = - y/x ×dy/dt- - - - - - - - - - 2
Substituting y = 5 into equation 1, it becomes
x² + 25 = 169
x² = 169 - 25 = 144
x = √144 = 12
Substituting x = 12, dy/dt = 3 and y = 5 into equation 2, it becomes
dx/dt = - 5/12 × 3
dx/dt = - 5/4 = - 1.25 ft/s
