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?

Answer 1

See the answer below:

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

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

  1. Formulate the problem as a optimization problem.
  2. Use the Pythagorean theorem to express the length of the ladder in terms of the distances involved.
  3. 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).
  4. Set the derivative equal to zero and solve for the critical point.
  5. Verify that the critical point corresponds to a minimum by checking the second derivative.
  6. 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} ]

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7