# A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft 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?

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To find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building:

- Formulate the problem as a optimization problem.
- Use the Pythagorean theorem to express the length of the ladder in terms of the distances involved.
- Differentiate the expression for the length of the ladder with respect to one of the variables (either the distance from the fence to the wall or the distance along the ground).
- Set the derivative equal to zero and solve for the critical point.
- Verify that the critical point corresponds to a minimum by checking the second derivative.
- Calculate the length of the ladder using the optimized distance found.

In this case:

Let ( x ) be the distance along the ground from the base of the building to the base of the ladder, and ( L ) be the length of the ladder.

Using the Pythagorean theorem, we have:

[ L^2 = (8^2 + x^2) ]

Differentiating both sides with respect to ( x ), we get:

[ \frac{dL}{dx} = \frac{1}{2} \cdot \frac{1}{\sqrt{8^2 + x^2}} \cdot 2x = \frac{x}{\sqrt{8^2 + x^2}} ]

Setting ( \frac{dL}{dx} ) equal to zero, we find the critical point:

[ \frac{x}{\sqrt{8^2 + x^2}} = 0 ]

Solving this equation, we get ( x = 0 ).

Since ( x = 0 ) corresponds to the distance from the building to the ladder's base, it doesn't make physical sense, so we discard it.

Instead, we consider the endpoints of the interval, where ( x = 4 ) (the distance from the fence to the wall).

Thus, the length of the shortest ladder is given by:

[ L = \sqrt{8^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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