Answer:
(a) shortest ladder length ≈ 35.7 ft (rounded to tenth)
(b) L = (d/x +1)√(16+x²) . . . . where 16 is fence height squared
Step-by-step explanation:
It works well to solve the second part of the problem first, then put in the specific numbers.
We have not been asked anything about "b", so we can basically ignore it. Using the Pythagorean theorem, we find the length GH in the attached drawing to be ...
GH = √(4²+x²) = √(16+x²)
Then using similar triangles, we can find the ladder length L to be that which satisfies ...
L/(d+x) = GH/x
L = (d +x)/x·√(16 +x²)
The derivative with respect to x, L', is ...
L' = (d+x)/√(16+x²) +√(16+x²)/x - (d+x)√(16+x²)/x²
Simplifying gives ...
L' = (x³ -16d)/(x²√(16+x²))
Our objective is to minimize L by making L' zero. (Of course, only the numerator needs to be considered.)
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(a) For d=24, we want ...
0 = x³ -24·16
x = 4·cuberoot(6) ≈ 7.268 . . . . . feet
Then L is
L = (24 +7.268)/7.268·√(16 +7.268²) ≈ 35.691 . . . feet
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(b) The objective function is the length of the ladder, L. We want to minimize it.
L = (d/x +1)√(16+x²)
