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

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.)

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Hailey 1 week 2021-09-14T23:39:55+00:00 1 Answer 0

Answers ( )

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    2021-09-14T23:41:37+00:00

    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

    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:

    \frac{dh}{dt}=-0.2 m/s

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

    \frac{dx}{dt}=0.2 m/s

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

    2x\frac{dx}{dt}+2h\frac{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=\sqrt{44.89/2} \\$x=4.738\:meters\\Therefore:\\h=4.738\:meters

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45:7+7-4:2-5:5*4+35:2 =? ( )