Answer:
x=4.738 meters
h=4.738 meters
Step-by-step explanation:
a = length of the ladder.
h = height of the ladder’s top at time t, and
x = distance from the wall to the ladder’s bottom.
From Pythagoras Theorem
[tex]a^2=x^2+h^2[/tex]
If a=6.7 meters, then:
[tex]6.7^2=x^2+h^2[/tex]
The top is sliding down the wall(decreasing) at a rate of 0.2m/s, therefore:
[tex]\frac{dh}{dt}=-0.2 m/s[/tex]
If the top and bottom of the ladder move at the same speed, then:
[tex]\frac{dx}{dt}=0.2 m/s[/tex]
Taking derivative of [tex]a^2=x^2+h^2[/tex]
[tex]2x\frac{dx}{dt}+2h\frac{dh}{dt}=0[/tex]
[tex]2x(0.2)+2h(-0.2)=0\\0.4x=0.4h\\x=h[/tex]
From [tex]6.7^2=x^2+h^2[/tex]
Since x=h
[tex]6.7^2=x^2+x^2\\2x^2=6.7^2\\x^2=44.89/2\\x=\sqrt{44.89/2} \\$x=4.738\:meters\\Therefore:\\h=4.738\:meters[/tex]
