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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The board, fence and ground form a right special triangle. The angles are 30, 60 and 90. Since the board is 4 feet, it forms the hypotenuse of the triangle. The side opposite the 30 degrees (fence side) is 2 feet, or half of the hypotenuse. The side opposite the 60 degrees (ground side) is 2 times square root 3, which is approximately 3.464 feet.

The inscribed circle, or incircle, of the triangle is the largest diameter pipe that will fit. The equation for the radius of the circle is 2 times the area of the triangle divided by the perimeter of the triangle (2a/p).

In this problem, the area of the triangle is .5bh = .5(2)(2sqrt(3)) = 2sqrt(3).

The perimeter is 2+2sqrt(3)+4 = 6+2sqrt(3)

radius = 2(2sqrt(3))/(6+2sqrt(3)) = 6.928/9.464 = .732

The radius of the pipe is approximately .732 feet.

The diameter of the larget pipe is 2(.732) = **1.464 feet.**

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