Question

An 8 ft tall fence runs parallel to the wall of a house at a distance of 10 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house is 20 ft high and the horizontal ground extends 25 ft from the fence.

What is the length of the shortest ladder?

Answer 1

Answer:

25.40 m

Step-by-step explanation:

given,

height of fence = 8 ft

horizontal distance = 10 ft

In Δ PRT and Δ QRS

$\dfrac{b}{b+10}=\dfrac{8}{a}$

$a = \dfrac{8(b+10)}{b}$

using Pythagoras theorem

$L = \sqrt{a^2+(10+b)^2}$

$L = \sqrt{(\dfrac{8(b+10)}{b})^2+(10+b)^2}$

$L = \dfrac{b+10}{b}\sqrt{64+b^2}$

for extrema differentiating both side

l'(b) = 0

$\dfrac{b^3-640}{b^2\sqrt{b^2+64}}=0$

      b³- 640 = 0

     b= 4∛10

now,

$L = \dfrac{4\sqrt[3]{10}+10}{4\sqrt[3]{10}}\sqrt{64+(4\sqrt[3]{10})^2}$

    L = 25.40 m

hence, length of the shortest ladder is equal to L = 25.40 m