What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

Question

A fence 5 feet tall runs parallel to a tall building at a distance of 5 feet 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?

 

Length of ladder=? feet. 

Answer 1

This problem can also be solved using calculus techniques.

 

We need a figure similar to the one described in the non-calculus answer.   If we call h the height above the ground at which the ladder touches the building, and x the distance from the foot of the ladder to the fence, by similar triangles we have  h/(5 +x) =  5/x.

By the Pythagorean theorem we have    L² = h² + (5+x)²

We can eliminate h  to get 

 

L² = 25(5+x)²/x² + (5+x)²  

 

We want L to be a minimum, this implies that L²  must also be a minimum with respect to x. 

This is an optimization problem in calculus.

We take the derivative of the right hand side with respect to x and set it equal to zero. 

The derivative can be worked out analytically using a combination of the product rule and the chain rule.

The resulting equation in x  can be rewritten  (taking advantage of a nice cancellation) in the form

 

   x³ =  125       So x = 5.  

 

This result can be substituted back into the L² equation to get  L² =200  ,  L =  14.14

Another thing to notice is that with x = 5,   the angle that the ladder makes with the ground

must be 45deg.  So L is just the diagonal of a 45 45 90 triangle with side 10.    Thus L = 10 sqrt(2)

=  14.14.

 

 

Answer 2

Hi Jessica

 

You may want to draw a figure.  When the ladder is in place, it forms a right triangle. If you draw a vertical line from where the ladder touches the ground, and a horizontal line from where it touches the wall, they meet to form a rectangle enclosing the ladder/wall and fence figure.  

 

The ladder forms a diagonal of this rectangle.  The rectangle with the shortest diagonal is always a square.  This means that the distance from the fence to where the ladder hits the ground is equal to the distance from a point 5 feet up the wall to where the ladder meets the wall.

 

All this means that the triangle formed by the ladder and fence is congruent to the one formed at the top by the ladder, wall and a line from the top of the fence to the wall. Also, it means that all the angles are 45 degrees.  The sides of the two congruent triangles are 5 feet, 5 feet, and 7.07 feet.

 

So, the ladder is 2 x 7.07 = 14.14 feet long.