What is the diameter of the largest pipe which will fit between the board, the fence and the ground?A board leaning against a fence makes an angle of 30 degrees with the horizontal. If the board...

 What is the diameter of the largest pipe which will fit between the board, the fence and the ground?

A board leaning against a fence makes an angle of 30 degrees with the horizontal. If the board is 4 feet long, what is the diameter of the largest pipe which will fit between the board, the fence and the ground?

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embizze | High School Teacher | (Level 1) Educator Emeritus

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A board leaning against a fence makes an angle of 30 degrees with the horizontal. If the board is 4 feet long, what is the diameter of the largest pipe which will fit between the board, the fence and the ground?

We assume the pipe touches the ground, the fence, and the board. Since the board is 4 feet long, the fence is 2 feet high where the board touches it, and the distance from the bottom of the fence to the bottom of the board is `2sqrt(3)` feet, using the 30-60-90 right triangle.

Let the distance from the bottom of the board to the point the board touches the pipe be x feet, and the radius of the pipe be r feet. Then the distance from the bottom of the board to the point the pipe touches the ground is x feet (tangent segments drawn from a point to a circle are congruent), and the distance from where the pipe touches the ground to the base of the fence is r feet. (The radius is perpendicular to the fence and the ground, forming a square of side length r feet)

Thus `x+r=2sqrt(3)` . But we also have a right triangle formed by the bottom of the board, the point the pipe touches the ground, and the center of the circle with an acute angle of `15^circ` . Then `r/x=tan15^circ` or `r=xtan15^circ` .

Substituting we get `x+xtan15^circ = 2sqrt(3)` . Noting that `tan15^circ = 1/(2+sqrt(3))` (Using a half-angle formula) we now have:

`x(1+1/(2+sqrt(3)))=2sqrt(3)`
`x=(2sqrt(3))/(1+1/(2+sqrt(3)))`
`x=(4sqrt(3)+6)/(3+sqrt(3))` by rationalizing
`x=sqrt(3)+1`

 

Then `x+r=2sqrt(3) => r=2sqrt(3)-(sqrt(3)+1)=sqrt(3)-1`

Thus the diameter is twice r or `2sqrt(3)-2 ~~1.464` feet.

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