Question

fence 3 feet tall runs parallel to a tall building at a distance of 7 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

If the foot of the ladder is x feet from the fence, and it reaches to a height h on the wall, then using similar triangles,

x/3 = (x+7)/h
h = 3(x+7)/x

the ladder's length z is

z^2 = (x+7)^2 + h^2
= (x+7)^2 + (3(x+7)/x)^2

z = (x+7)√(x^2+9) / x

So, for minimum z, we need dz/dx = 0

dz/dx = (x^3-63) / x^2√(x^2+9)
dz/dx=0 when x=∛63, or about 4 feet

So, minimum z is roughly

(4+7)√(16+9)/4 = 55/4 = 13.75 ft

Answer 2

please i need help, oi was looking towards similar problem, but i'm not sure how cube route 75 came into play with one of the problems that had 5 feet tall running parallel at 3 ft distance

Answer 3

oh thank you. this formate shows a lot better. Thank you Steve.

Answer 4

in that case,

z^2 = (x+3)^2 + (5(x+3)/x)^2
z = (x+3)√(x^2+25) / x

dz/dx = (x^3-75) / x^2√(x^2+25)

and x = ∛75