How fast is the top of the ladder sliding down the wall?

A 20ft ladder is leaning against a wall and begins to slide. How fast is the top of the ladder sliding down the wall as the instant of time when the bottom of the ladder is 12ft from the wall and is sliding away from the wall at a rate of 5ft/sec.

2 Answers

  • Amy
    Lv 7
    1 month ago
    Favourite answer

    Use the Pythagorean theorem to write the position of the top of the ladder as a function of the position of the bottom of the ladder.

    Take the derivative of this function with respect to time. 

    Your result will include x and dx/dt, which were given in the description.

  • Ash
    Lv 7
    1 month ago

    The ladder forms a right angles triangle with the wall.

    let b = base of ladder from wall

    and h = height ladder from ground

    we have, b² + h² = 20² ......(1)

    h² = 400 - b²

    h = √(400-b²)

    Derive (1) with respect to time

    2b db/dt + 2h dh/dt = 0

    2b db/dt + 2√(400-b²) dh/dt = 0

    b db/dt + √(400-b²) dh/dt = 0

    dh/dt = -(b db/dt) /√(400-b²)

    dh/dt = -(12*5) /√(400-12²)

    dh/dt =-3.75 ft/s    ← rate at which top of the ladder is sliding down the wall

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