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Answer: 13 meters</h3>
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Explanation:
L = length of ladder
Draw an xy axis. Focus on the upper right quadrant where x > 0 and y > 0
Let the origin (0,0) be the base of the wall.
Let y be the height of the top of the ladder and x be the horizontal distance the base of the ladder is from the base of the wall
Since the ladder is sliding down at a rate of 0.25 m/s, this means dy/dt = -0.25
At x = 5, we are given that dx/dt = 0.6
Use the pythagorean theorem to see that
a^2+b^2 = c^2
x^2 + y^2 = L^2
Now apply the derivative to both sides with respect to t. Don't forget to apply the chain rule while doing so.
d/dt[ x^2 + y^2 ] = d/dt[ L^2 ]
d/dt[ x^2 ] + d/dt[ y^2 ] = d/dt[ L^2 ]
2x*dx/dt + 2y*dy/dt = 0
2(x*dx/dt + y*dy/dt) = 0
x*dx/dt + y*dy/dt = 0
Note how the right hand side goes to 0 because the derivative of a constant is 0. The ladder length L doesn't change, so neither does L^2.
From here we plug in x = 5, dx/dt = 0.6, dy/dt = -0.25 and we solve for y
x*dx/dt + y*dy/dt = 0
5*0.6 + y*(-0.25) = 0
0.3 - 0.25y = 0
3 = 0.25y
0.25y = 3
y = 3/0.25
y = 12
The top of the ladder is y = 12 meters off the ground and the base is x = 5 meters from the wall.
Therefore,
L^2 = x^2 + y^2
L = sqrt(x^2 + y^2)
L = sqrt(5^2 + 12^2)
L = 13
The ladder is 13 meters long.
