Question

Upper AA 66​-ft-tall fence runs parallel to a wall of a house at a distance of 3030 ft. Find the length of the shortest ladder that extends from the​ ground, over the​ fence, to the house. Assume the vertical wall of the house and the horizontal ground have infinite extent.

Answer 1

46.64 ft

Explanation:

From the diagram, triangle ACE is similar to triangle BCD.

Therefore:

`(b)/(30+b) = (6)/(a)`

`a=(6(30+b))/(b)`

Let the Length of the Ladder, L

Using Pythagoras' theorem:

`L^(2)=(30+b)^(2)+\left((6(30+b))/(b)\right)^(2)\\ \\L^(2)=((30+b)^(2)(36+b^(2)))/(b^(2))\\\\ L=\sqrt{((30+b)^(2)(36+b^(2)))/(b^(2)) }\\\\L=(30+b)/(b)√(b^2+36)`

To find the least value, we have to equate the derivative of L to 0.

`L^(\prime)=(b^(3)-1080)/(b^(2)√(b^2+36))`

Therefore:

`b^3-1080=0\\\\b^3=1080\\\\b=10.26`

Substitute b=10.26 to find L:

`L=(30+b)/(b)√(b^2+36)\\\\ L=(30+10.26)/(10.26)√(10.26^2+36)\\\\L\approx 46.64\;ft`

The length of the shortest ladder that extends from the​ ground, over the​ fence, to the house is 46.64 feet.