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

Answer:

0.33 ft

Step-by-step explanation:

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

Therefore,

$\frac{b}{30+b} = \frac{6}{a}$

$a=\frac{6(30+b)}{b}$

Let the Length of the Ladder, L

Using Pythagoras theorem

$L^{2}=(30+b)^{2}+(\frac{6(30+b)}{b})^{2}$

$L^{2}=\frac{(30+b)^{2}(36+b^{2})}{b^{2}}$

$L=\sqrt{\frac{(30+b)^{2}(36+b^{2})}{b^{2}} }$

$L=\frac{30+b}{b}\sqrt{b^2+36}$

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

$L^{1}=\frac{b^{3}-1080}{b^{2}\sqrt{b^2+36}}$

Therefore:

b³-1080=0

b³=1080

b=10.26

The minimum length of the ladder is:

$L=\frac{30+10.26}{10.26}\sqrt{10.26^2+36}$

=0.33 ft