applications of derivative problem

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PaulKraemer

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Apr 10, 2011
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Hi,

I have a problem that goes like this: "A fence 8 feet tall stands on level ground and runs parallel to a tall building. If the fence is 1 foot from the building, find the shortest ladder that will extend from the ground over the fence to the wall of the building."

If x is the vertical distance from the top of the fence to where the ladder touches the building, I used similar triangles to determine that the horizonal distance from the bottom of the fence to where the ladder touches the ground would be 8/x.

I used the pythagorean theorem to determine that the length L of the ladder would be:

L(x) = sqrt [ (8/x + 1)^2 + (8 + x^2) ]

If the above formula is correct, I believe I have to find the derivative L'(x), find the critical numbers, and see if one of the critical numbers is a minimum.

When I try to take the derivative of the above function L(x), I keep getting a really complicated formula that I cannot find any critical numbers for. I was wondering if anyone could tell me (1) is the formula I came up with for L(x) correct? and if so (2) Is there a trick that would make finding this derivative easier? This is only question #5 in the chapter I'm on, so I wouldn't think it should be so difficult.

Thanks in advance,
Paul
 
PaulKraemer said:
Hi,

I have a problem that goes like this: "A fence 8 feet tall stands on level ground and runs parallel to a tall building. If the fence is 1 foot from the building, find the shortest ladder that will extend from the ground over the fence to the wall of the building."

If x is the vertical distance from the top of the fence to where the ladder touches the building, I used similar triangles to determine that the horizonal distance from the bottom of the fence to where the ladder touches the ground would be 8/x.

I used the pythagorean theorem to determine that the length L of the ladder would be:

L(x) = sqrt [ (8/x + 1)^2 + (8 + x^2) ]

If the above formula is correct, I believe I have to find the derivative L'(x), find the critical numbers, and see if one of the critical numbers is a minimum.

When I try to take the derivative of the above function L(x), I keep getting a really complicated formula that I cannot find any critical numbers for. I was wondering if anyone could tell me (1) is the formula I came up with for L(x) correct? and if so (2) Is there a trick that would make finding this derivative easier? This is only question #5 in the chapter I'm on, so I wouldn't think it should be so difficult.

Thanks in advance,
Paul
Click to expand...

L(x) = sqrt [ (8/x + 1)^2 + (8 + x)^2] ............................... I am assuming that was typo

L^2 = (8/x + 1)^2 + (8 + x)^2

L * L' = -16(8/x +1)/x^2 + 2(8+x)

8(8/x + 1) = x^2(8+x)

8(8+x) = x^3(8+x)

x = 2
 
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