Answer:
10.81 ft
Explanation:
You want the length of the shortest ladder that will reach over a 2 ft high fence to reach a building 6 ft from the fence.
Trig relations
Relevant trig relations are ...
Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Ladder length
In the attached diagram, the ladder is show as line segment CD, intersecting the top of the fence at point B. The length of the ladder is the sum of segment lengths BD and CB.
Using the above trig relations, we can write expressions that let us find these lengths in terms of the angle at D:
sin(D) = AB/BD ⇒ BD = AB/sin(D)
cos(D) = BG/CB ⇒ CB = BG/cos(D)
Then the ladder length is ...
CD = BC +CB = AB/sin(D) +BG/cos(D)
CD = 2/sin(D) +6/cos(D)
Minimum
The minimum can be found by differentiating the length with respect to the angle. This lets us find the angle that gives the minimum length.
CD' = -2cos(D)/sin²(D) +6sin(D)/cos²(D)
CD' = 0 = (6sin³(D) -2cos³(D))/(sin²(D)cos²(D)) . . . common denominator
0 = 3sin³(D) -cos³(D) . . . . the numerator must be zero
Factoring the difference of cubes, we have ...
0 = (∛3·sin(D) -cos(D))·(∛9·sin²(D) +∛3·sin(D)cos(D) +cos²(D))
The second factor is always positive, so the value of D can be found from
∛3·sin(D) = cos(D)
D = arctan(1/∛3) . . . . . . . divide by ∛3·cos(D), take inverse tangent
D ≈ 37.736°
CD = 2/sin(37.736°) +6/cos(37.736°) = 3.51 +7.30 = 10.81 . . . feet
The shortest ladder that reaches over the fence to the building is 10.81 feet.
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Additional comments
The second attachment shows a graphing calculator solution to finding the minimum of the length versus angle in degrees.
The ladder length can also be found in terms of the distance AD.
L = BD(1 +BG/AD) = (1 +6/AD)√(4+AD²)
The minimum L is found when AD=∛(BG·AD²) = ∛24 ≈ 2.884.