Calculus

"A 9-foot wall stands 30 ft from the building. find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall".

So I believe I need to start with the Pythagorean Theorem formula, but where do I need to go from there? All help is appreciated!

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  1. I would use trig.
    Make a sketch, showing the ladder touching the top of the fence.
    I see 2 similar right-angled triangles, both with a base angle of θ.
    Let the hypotenuse of the triangle touching the wall be L1
    Let the hypotenuse of the triangle touching the ground be L2

    for L1, cosθ = 30/L1 ----> L1 = 30/cosθ
    for L2, sinθ = 9/L2 ----> L2 = 9/sinθ

    Let L = L1 + L2
    = 30/cosθ + 9/sinθ = 30(cosθ)^-1 + 9(sinθ)^-1
    dL/dθ = -30(cosθ)^-2 (-sinθ) - 9(sinθ)^-2 (cosθ)
    = 30sinθ/cos^2 θ - 9cosθ/sin2 θ
    = 0 for a min of L

    30sinθ/cos^2 θ = 9cosθ/sin2 θ
    30 sin^3 θ = 9cos^3 θ
    sin^3 θ/cos^3 θ = 3/10 = .3
    tan^3 θ = .3
    tanθ = (.3)^(1/3) = .66943...
    θ = 33.7996... °

    then L1 = 30/cos 33.799.. ° = 36.1 ft
    and L2 = 9/sin 33.799.. ° = 16.2 ft

    L = 36.1 + 16.2 ft = 52.3 ft

    Phewww!!!!
    check my calculations please

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    Reiny
  2. Thank you so much, Mr. Reiny, you are amazing!

    I checked the calculations and they were correct. 52.3 ft was one of the answer choices!

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  3. Or, if you want to use the Pythagorean Theorem,
    L = length of beam
    x = distance from wall to the foot of the beam
    h = height where beam touches the wall
    Then, using similar triangles,
    h/(x+30) = 9/x
    h = 9(x+30)/x
    L^2 = (x+30)^2 + h^2 = (x+30)^2 + 81((x+30)/x)^2
    dL/dx = 0 when x = ∛2470 ≈ 13.44
    h = 9(13.44+30)/13.44 = 29.08
    Thus, L^2 = 43.44^2 + 29.08^2
    L = 52.27
    which agrees with Reiny's result >whew<

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