The top of a ladder slides down a vertical wall at a rate of 0.675 m/s. At the moment when the bottom of the ladder is 6 m from the wall, it slides away from the wall at a rate of 0.9 m/s. How long is the ladder?

Answer:The length of the ladder is 10 m.Step-by-step explanation:Let x shows the distance of the top of ladder from the bottom of base of the wall, y shows the distance of the bottom of ladder from the base of the wall and l is the length of the ladder,Given,[tex]\frac{dx}{dt}=-0.675\text{ m/s}[/tex][tex]\frac{dy}{dt}=0.9\text{ m/s}[/tex] y = 6 m,Since, the wall is assumed perpendicular to the ground,By the pythagoras theorem,[tex]l^2=x^2+y^2[/tex]Differentiating with respect to t ( time ),[tex]0=2x\frac{dx}{dt}+2y\frac{dy}{dt}[/tex]     ( the length of wall would be constant )By substituting the value,[tex]0=2x(-0.675)+2(6)(0.9)[/tex][tex]0=-1.35x+10.8[/tex][tex]\implies x=\frac{10.8}{1.35}=8[/tex]Hence, the length of the ladder is,[tex]L=\sqrt{x^2+y^2}=\sqrt{8^2+6^2}=\sqrt{64+36}=\sqrt{100}=10\text{ m}[/tex]