How would I solve this?

Question

A fence 3 feet tall runs parallel to a tall building at a distance of 5 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?

I don't have the hegiht of the building so there must be another way to solve this problem without it?

Answer 1

h / (x + 5) = 3/x
L^2 = h^2 + (x + 5)^2
h = 3(x + 5)/x
L^2 = 9(x + 5)^2/x^2 + (x + 5)^2
L^2 = (x + 5)^2(9 + x^2) / x^2
2LdL = [- 2(x + 5)^2(9 + x^2) / x^3 + 2x(x + 5)^2 / x^2 + 2(x + 5)(9 + x^2) / x^2]dx
dL/dx = [- (x + 5)^2(9 + x^2) / x^3 + x(x + 5)^2 / x^2 + (x + 5)(9 + x^2) / x^2]x^2 / (x + 5)^2(9 + x^2)
dL/dx = [- (x + 5)(9 + x^2) + x^2(x + 5) + x(9 + x^2)] / x(x + 5)(9 + x^2) = 0 for min/max
[- (x + 5)(9 + x^2) + x^2(x + 5) + x(9 + x^2)] = 0
(x + 5)(x^2 - 9 - x^2) + x(9 + x^2) =0
x^3 + 9x -9x - 45 = 0
x^3 = 45
x ≈ 3.556893
L^2 = (x + 5)^2(9 + x^2) / x^2
L^2 ≈ (73.22042)(21.65149) / 12.65149
L^2 ≈ 125.3079
L ≈ 11.19410 ft. ≈ 11 ft. 3 in.
for shortest ladder in feet, round up to 12 ft.

Answer 2

the 2nd reply misunderstood the qn.

the 3rd reply said that the min distance between the building and the fence is 4. that is wrong too since you mentioned that the fence is parallel to the building at a distance of 5ft.

i think you would need to use the similar triangles property. assuming that an isosceles triangle will have the shortest hypotenuse, the base of the ladder to the fence will be at a distance of 3ft, because tan45 = 3/x.

the overall length from the point the ladder touches the ground will hence be 3 + 5. Since we are using the isosceles triangle, the point where the ladder touches the building will be 8ft above ground as well.

thus, the length of the shortest ladder will be sqrt(8^2 + 8^2) using pythagoras theorem, which is 11.3 ft.

Answer 3

It would be best if the height of the building is provided... However... This is what i think ^^

From the top of the fence (3ft) with a distance of 5ft to the building, the minimum distance to the building would be 4ft (Using theorem pythagoras method). This will gives the ladder with a slope of 3/5. Ladder is a linear product, hence the slope must be 3/5 throughout the ladder. Using this concept, another 4ft will be needed to reach the top fence from the ground.

Finally, it would be 4ft + 4ft = 8ft ladder ^^

Answer 4

You might want to add the height of the building... The taller the building, the closer to the wall the ladder can be.