Question

A fence 6 feet tall runs parallel to a tall building at a distance of 6 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 0.82 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. equation editorEquation Editor The distance along the ladder from the top of the fence to the wall is equation editorEquation Editor Using these hints write a function L(x) which gives the total length of a ladder which touches the ground at an angle x, touches the top of the fence and just reaches the wall. L(x) = equation editorEquation Editor . Use this function to find the length of the shortest ladder which will clear the fence. The length of the shortest ladder is equation editorEquation Editor feet.

Answer 1

Answer:

  12√2 feet ≈ 16.97 feet

Step-by-step explanation:

For the dimensions shown in the attached diagram, the distance "a" along the ladder from the ground to the fence is ...

  a = (6 ft)/sin(x) = (6 ft)/sin(0.82) ≈ 8.206 ft

The distance along the ladder from the top of the fence to the wall is ...

  b = (6 ft)/cos(x) = (6 ft)/cos(0.82) ≈ 8.795 ft

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In general, the distance along the ladder from the ground to the wall is ...

  L(x) = a +b

  L(x) = 6/sin(x) +6/cos(x)

This distance will be shortest for the case where the derivative with respect to x is zero.

  L'(x) = 6(-cos(x)/sin(x)² +sin(x)/cos(x)²) = 6(sin(x)³ -cos(x)³)/(sin(x)²cos(x²))

This will be zero when the numerator is zero:

  0 = 6(sin(x) -cos(x))(1 -sin(x)cos(x))

The last factor is always positive, so the solution here is ...

  sin(x) = cos(x)   ⇒   x = π/4

And the length of the shortest ladder is ...

  L(π/4) = 6√2 + 6√2

  L(π/4) = 12√2 . . . . feet

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The ladder length for the "trial" angle of 0.82 radians was ...

  8.206 +8.795 = 17.001 . . . ft

The actual shortest ladder is ...

  12√2 = 16.971 . . . feet