A ladder 5m long is standing vertically, flat against a vertical wall, while its lower end in on the horizontal floor. The lower end moves horizontally away from the wall at a constant speed of 1m/s while the upper end stays in contact with the wall. Find the speed at which the upper end is moving down the wall 4 seconds after the lower end has left the wall.
If y is the height and x is the distance from the wall, then
x^2+y^2=25
x dx/dt + y dy/dt = 0
After 4 seconds, x=4 and y=3 so,
4 * 1 + 3 dy/dt = 0
dy/dt = -4/3
so the top is sliding down the wall at 4/3 m/s
To find the speed at which the upper end of the ladder is moving down the wall, we can use similar triangles.
Let's label the distance from the bottom of the ladder to the wall as "x" and the distance from the top of the ladder to the floor as "h". From the problem statement, we know that x is changing at a constant rate of 1m/s.
Using the Pythagorean theorem, we can relate x, h, and the length of the ladder (5m):
x^2 + h^2 = 5^2
Differentiating both sides of this equation with respect to time (t), we get:
2x(dx/dt) + 2h(dh/dt) = 0
Since the lower end of the ladder is moving horizontally away from the wall at a constant speed of 1m/s, dx/dt = 1.
At t = 4 seconds, we need to find dh/dt.
Plugging in the known values, we have:
2x(dx/dt) + 2h(dh/dt) = 0
2(4)(1) + 2h(dh/dt) = 0 (substituting dx/dt = 1 and t = 4)
8 + 2h(dh/dt) = 0
Since the upper end of the ladder stays in contact with the wall, h is also changing. To find h, we can use the Pythagorean theorem again. At t = 4 seconds, x will be 4 meters (found by multiplying the elapsed time by the rate at which the lower end is moving away from the wall).
Substituting the value of x in the equation x^2 + h^2 = 5^2:
4^2 + h^2 = 5^2
16 + h^2 = 25
h^2 = 25 - 16
h^2 = 9
h = 3
Substituting the value of h in the equation 8 + 2h(dh/dt) = 0:
8 + 2(3)(dh/dt) = 0
8 + 6(dh/dt) = 0
6(dh/dt) = -8
dh/dt = -8/6
dh/dt = -4/3
Therefore, the speed at which the upper end of the ladder is moving down the wall 4 seconds after the lower end has left the wall is -4/3 m/s. The negative sign indicates that the upper end is moving downward.