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Calculus

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A fence 3 feet tall runs parallel to a tall building at a distance of 3 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.
Answer the following:
The distance along the ladder to the top of the fence is=?

Write a function L(x) which gives the total length of a ladder which touches the ground at an angle x, touches the top of the fence and just reaches the wall=?
The length of the shortest ladder is=?

  • Calculus -

    If the foot of the ladder is d feet from the fence,

    3/d = tan x
    d = 3cot x

    L(x) = (d+3) sec x
    = 3(1+cotx)secx

    dL/dx = 3secx(tanx - csc^2(x)+1)
    So, either secx=0 (no solution), or
    tanx - csc^2(x)+1 = 0
    x = π/4

    So, L(π/4) = 3(2)(√2) = 6√2

    As expected, a square has the shortest diagonal.

  • Calculus -

    Correction: If the ladder makes an angle 1.25 radians with the ground, touches the top of the fence, and just reaches the wall, the distance along the ladder from the ground to the top of the fence=?
    The distance along the ladder from the top of the fence to the wall is=?

  • Calculus -

    come on, man.
    distance to top of fence:

    d/3 = csc(1.25)

    You have the formula for L(x), the length of the ladder, so subtract d from that.

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