A fence 12ft high runs parallel to a tall building at a distance of 2ft 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?
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.
Let's call the distance from the ground to the top of the building "h" and the length of the ladder "L".
According to the problem, the fence is 12ft high and is 2ft away from the building. This forms a right-angled triangle. The height of the triangle is 12ft, the base is 2ft, and the hypotenuse is the length of the ladder L.
Using the Pythagorean theorem, which states that the sum of the squares of the lengths of the two shorter sides of a right triangle is equal to the square of the length of the hypotenuse, we have:
L^2 = 2^2 + 12^2
L^2 = 4 + 144
L^2 = 148
Taking the square root of both sides, we get:
L = √148
L ≈ 12.17ft
Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately 12.17ft.
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 sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
In this case, the ladder represents the hypotenuse, and the distances from the ground to the top of the fence and from the top of the fence to the building represent the two legs.
Let's denote the distance from the ground to the top of the fence as "a" and the distance from the top of the fence to the building as "b."
According to the problem, the height of the fence is 12ft, so a = 12ft. The distance from the fence to the building is given as 2ft, so b = 2ft.
Now, we can use the Pythagorean theorem, which is:
a^2 + b^2 = c^2
Substituting the given values, we have:
12^2 + 2^2 = c^2
144 + 4 = c^2
148 = c^2
To find the length of the ladder (c), we take the square root of both sides of the equation:
c = sqrt(148)
c ≈ 12.165ft
Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately 12.165ft.
If the ladder of length z touches the ground x feet from the fence, and touches the building at height y, then
z^2 = (x+2)^2 + y^2
Using similar triangles we see that
x/12 = (x+2)/y
y = 12(x+2)/x
z^2 = (x+2)^2 + (12(x+2)/x)^2
So, find dz/dx and set it zero, to see that the minimum z occurs at x = ∛288. So, the shortest ladder is 17.85 feet.