Question

A fence 2 feet tall runs parallel to a tall building at a distance of 6 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?

Answer 1

8feet

Answer 2

its not 8 feet

Answer 3

Let theta be the acute angle that the ladder makes with the ground. If you draw the diagram for this situation, you will see that the top of the fence divides the ladder into two pieces. Draw a horizontal line from the top of the fence to the building. Let x be the length of the lower piece and y be the length of the upper piece. Each piece is the hypotenuse of a right triangle, and those two triangles are similar. You can read from the right triangles that
\[L( \theta) = x + y = 2 \csc \theta + 6 \sec \theta.\]
\[dL/d \theta= -2 \csc \theta + 6 \sec \theta \tan \theta = 0\]
\[\csc \theta \cot \theta = 3\sec \theta \tan \theta\]
\[\cos \theta/\sin ^{2}\theta = 3\sin \theta/\cos ^{2}\theta\]
\[\tan \theta = 1/\sqrt[3]{3}\]

Answer 4

The length of the lower triangle's horizontal leg is
\[\sqrt{x ^{2}-4},\]
and the length of the upper triangle's vertical leg is
\[\sqrt{y ^{2}-36}.\]

Answer 5

Lower triangle:
\[\tan \theta = 2/ \sqrt{x ^{2}-4} = 1/ \sqrt[3]{3},\]
which solves to
\[x = 2 \sqrt{1 + \sqrt[3]{9}} \approx 3.51.\]

Answer 6

A similar calculation gives
\[y = 2 \sqrt{9 + 3 \sqrt[3]{3}}\approx 9.38.\]

Answer 7

Then
\[L = x + y \approx 12.9 ft.\]

Answer 8

Quite a project getting this one posted due to getting booted off the internet repeatedly.