A fence 6 feet tall runs parallel to a tall building at a distance of 2 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?
Make a diagram
let the foot of the ladder be x ft from the fence
let the ladder reach y ft above the ground
I see similar triangle so set up a ratio
6/x = y/(x+2)
xy = 6x+12
y = (6x+12)/x
let the length of the ladder be L
L^2 = (x+2)^2 + y^2
= (x+2)^2 + [(6x+12)/x]^2
2L dL/dx = 2(x+2) + 2[(6x+12)/x] (x(6) - (6x+12))/x^2
= 0 for a min of L
x+2 + (6x+12)(-12)/x^3 = 0
.. ..
x^4 + 2x^3 - 72x - 144 = 0
x^3(x+2) - 72(x+2) = 0
(x^3 - 72)(x+2) = 0
x = -2 , not possible
or
x = 72^(1/3) , (which is the cuberoot of 72)
sub back into L^2 = ....
THANKS YOU Reiny!!! You are a great help. Turns out my "y" was wrong but I did get the "x"
Did you get a shortest ladder of 9.81 ft ?
I actually got 10.811 ft
To find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the ladder is the hypotenuse, and the fence and the distance from the fence to the building are the other two sides.
Let's denote the length of the ladder as 'L'. According to the given information, the height of the fence is 6 feet, and the distance from the fence to the building is 2 feet.
Using the Pythagorean theorem, we can set up the equation:
L^2 = 6^2 + 2^2
Simplifying this equation, we get:
L^2 = 36 + 4
L^2 = 40
To find the length of the ladder, we need to take the square root of both sides of the equation:
L = √40
L ≈ 6.32 feet
Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately 6.32 feet.