A fence 3 feet tall runs parallel to a tall building at a distance of 3 feet fro

Question

A fence 3 feet tall runs parallel to a tall building at a distance of 3 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building. Here are some hints for finding a solution: Use the angle that the ladder makes with the ground to define the position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of the fence. If the ladder makes an angle 1.25 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the ground to the top of the fence. The distance along the ladder from the top of the fence to the wall is __________________. Using these hints write a function which gives the total length of a ladder which touches the ground at an angle , touches the top of the fence and just reaches the wall. L(x) = ____________. Use this function to find the length of the shortest ladder which will clear the fence. The length of the shortest ladder is feet.

Explanation / Answer

You want the length of the ladder as a function of the angle (alpha) it forms with the ground. This is: L=(a+b/tan(alpha))/cos(alpha) where a is 7' and b 6'. Differentiating against alpha and equating to zero gives you the optimal value for the angle: dL/d(alpha)=(a*tan^2(alpha)-b*cot(alph… alpha=tan^-1((b/a)^(1/4)) = 43.9 degrees = 0.766 rad Put this back in L: L=18' 4"=220" If the ladder makes 1.31 radians (beta) with the ground and touches at the same time ground, fence and wall: L=(a+b*cot(beta))/cos(beta)=33' 3"=400" And from the fence to the wall is: D=L-b/sin(beta)=26' 1"=313"

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