Suppose that the ladder rests on the ground distance x from the fence on the side away from the building. Then the length of the ladder from the ground to the top of the fence is
l1 = √(3² + x²)
The length of the ladder from the top of the fence to the building is proportional to that:
l2 = l1×(2/x)
Then the total length of the ladder in terms of x is
l(x) = l1 + l2 = (1 +2/x)√(9+x²)
To find the value of x that makes this be a minimum, we can differentiate and set the derivative to zero.
dl/dx = (-2/x²)√(9+x²) +(1+2/x)(2x)/(2√(9+x²))
0 = -2√(9+x²)/x² + (x+2)/√(9+x²)
0 = -2(9+x²) + x³ +2x²
x³ = 18
x = ∛18 ≈ 2.6207
So, the length of the ladder is
l(∛18) = (1 + 2/∛18)√(9+(∛18)²) ≈ 7.0235 . . . . feet
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The general solution is somewhat interesting.
If we define
α = the angle the ladder makes with the ground
x = fence distance from building
y = fence height
then we find that
α = arctan(∛(y/x))
and the length of the ladder is
ladder length = y/sin(α) + x/cos(α)
In this problem, that means
α = arctan(∛(3/2)) ≈ 48.86°
ladder length = 3/sin(48.86°) + 2/cos(48.86°)
= 3.9835 + 3.0400 = 7.0235 . . . . same as above
I find the tangent of the ladder angle being the cube root of the tangent of the angle from the building to the fence top to be an interesting relationship. No matter the height of the fence in relation to its distance from the building, the ladder angle is closer to 45° than is the angle to the fence top.
