Solve Calculus Problem: Find Shortest Ladder Length

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In summary, the problem asks for the shortest ladder needed to reach a tall building from the other side of a 6ft high fence that is 2ft away from the building. By expressing the length of the ladder as a function of either the height or distance, and differentiating with respect to that variable, the minimum length of the ladder can be found. The key is finding an extremum of L^2, which is simpler to differentiate. The conversation discusses the process and suggests using the quotient rule for the second term.
  • #1
AD
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Not my own homework problem, but somebody else's.

Parallel to a tall building runs a 6ft high fence. It is 2ft away from the building. What is the shortest ladder needed to reach the building from the other side of the fence?

I've drawn a diagram for this problem (see attached).

I've called the height above the ground at which the ladder touches the building 'h' and the distance of the base of the ladder from the fence 'x' and the length of the ladder 'L'.

Then

L2 = h2 + (x + 2)2

I think that by expressing L as a function in terms of one of the variables, x or h, and differentiating with respect to that variable, I can find then find the minimum length of the ladder.

By considering similar triangles

6/x = h/(2 + x)

So

x = 12/(h - 6)

And

h = 12/x + 6

But by substituting in either of these values, the differentiation dL/dx or dL/dh becomes horrible.

Is there an easier way to solve this problem? Am I overlooking something? I'd like to see if I could find an easier function to differentiate.
 

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  • #2
This is the trick: Look for an extremum of L2. You know that L is positive, so an extremum of L2 is also an extremum of L.
I've tried
[tex]
\frac{d}{dh}L^2 = 0
[/tex]
It's lengthy but easy.

Edit: I think the rest of your work is OK.
 
Last edited:
  • #3
Okay, so I can do dL2/dx or dL2/dh rather than and dL/dx or dL/dh and that will make the maths easier.

You've suggested dL2/dh.

Substituting x = 12/(h - 6) into the equation for L2 means that I would have to differentiate

h2 + [(12/(h - 6)) + 2]2

with respect to h. Right?
 
  • #4
Correct.
Simplify before you derive. A lot of things will cancel.:wink:
 
  • #5
So

L2 = h2 + [(12/(h - 6)) + 2]2

= h2 + 144/[(h - 6)2] + 48/(h - 6) + 4

What's cancelling?
 
  • #6
Oh, c'mon. Combine the fractions before you square!
 
  • #7
To get

L2 = h2 + 4h2/(h - 6)2

...?

Can I simplify it further?
 
  • #8
I think this will do. Next, derive...
 
  • #9
How? Using the quotient rule for the second term? Is that what you did?
 
  • #10
Yes.
 

What is a calculus problem?

A calculus problem involves using mathematical principles and techniques, particularly those of calculus, to solve a specific problem or scenario. It often involves finding the optimal solution to a given situation.

What is the shortest ladder length problem?

The shortest ladder length problem is a classic optimization problem in calculus. It involves finding the shortest possible length of a ladder that can reach from the ground to a specific point on a vertical wall, given the distance between the wall and the base of the ladder.

What are the steps to solve a shortest ladder length problem?

To solve a shortest ladder length problem, you must first identify the variables involved, such as the distance from the wall to the base of the ladder and the height of the wall. Then, you can use calculus techniques, such as differentiation and optimization, to find the minimum length of the ladder that satisfies the given constraints.

What are some real-world applications of the shortest ladder length problem?

The shortest ladder length problem has many real-world applications, such as determining the minimum length of a ladder needed to reach a specific height on a building or the minimum length of a ramp needed to accommodate a wheelchair. It is also used in engineering and construction to design structures that can safely reach certain heights.

Are there any tips for solving a shortest ladder length problem?

Yes, some tips for solving a shortest ladder length problem include drawing a diagram to visualize the problem, identifying the variables and their relationships, and using calculus techniques such as differentiation and optimization to find the optimal solution. It can also be helpful to check your answer by plugging it back into the original problem to ensure it satisfies all given constraints.

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