Question

The variable a is the length of the ladder. The variable h is the height of the ladder’s top at time t, and x is the distance from the wall to the ladder’s bottom. Suppose that the length of the ladder is 6.7 meters and the top is sliding down the wall at a rate of 0.2m/s. What are the values of h and x at the moment when the top and bottom of the ladder move at the same speed? (Use decimal notation. Give your answer to three decimal places.)

Answer 1

x=4.738 meters

h=4.738 meters

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

`a^2=x^2+h^2`

If a=6.7 meters, then:

`6.7^2=x^2+h^2`

The top is sliding down the wall(decreasing) at a rate of 0.2m/s, therefore:

`(dh)/(dt)=-0.2 m/s`

If the top and bottom of the ladder move at the same speed, then:

`(dx)/(dt)=0.2 m/s`

Taking derivative of
`a^2=x^2+h^2`

`2x(dx)/(dt)+2h(dh)/(dt)=0`

`2x(0.2)+2h(-0.2)=0\\0.4x=0.4h\\x=h`

From
`6.7^2=x^2+h^2`

Since x=h

`6.7^2=x^2+x^2\\2x^2=6.7^2\\x^2=44.89/2\\x=√(44.89/2) \\$x=4.738\:meters\\Therefore:\\h=4.738\:meters`