INGE 3031 Statics Workbook WB:25 3.2. A contractor is hired to prepare a soil su

Question

INGE 3031 Statics Workbook WB:25 3.2. A contractor is hired to prepare a soil surface so that it is flat. Parameter Values 3.2 ft t 20% he expected standard is that the finished surface is approximately a plane that is not inclined by more than 0.5° in any direction. After leveling the and with machinery, the contractor performs a survey to determine if the land is sufficiently level. The measured elevation ofpoints A and B. relative to the elevation of point C, are A, and B, respectively 00 ft 505 ft 400 ft 500 ft = In the figure above, the points A and B appear to be located exactly on the x- and y- aves, respectively, even though the are slightly below. Explain why the diagram appears to be "wrong" tn heln determine how "flat" o

Explanation / Answer

Given, Az = -3.2 ft
Ax = 400 ft
Bz = -4.4 ft
By = 505 ft

so, vector AC = |Ax|(-sqroot(1 - [Az/|Ax|]^2)i - [Az/|Ax|] k) = 400(-sqroot(1 - (3.2/400)^2 i - 3.2K/400)) = -399.98i - 3.2k
vector CB = |By|(sqroot(1 - [Bz/|By|]^2)j - [Bz/|By|]k) = 505(sqroot(1 - 4.4/505)j - 4.4/505 k) = 502.79j - 4.4k
vector BA = sqroot(AC^2 + CB^2)[(Ax)i - Byj + (Az - Bz)k] / |[(Ax)i - Byj + (Az - Bz)k]| = sqroot(400^2 + 505^2)[(400)i - 505j + (-3.2 + 4.4)k] / |[(Ax)i - Byj + (Az - Bz)k]|
BA = 0.999(400i - 505j + 1.2 k)
BA = 399.6i - 504.495j + 1.1988k

now, AC + CB = -399.98i + 502.79j - 7.6 k = -BA = -399.6i + 504.495j + 1.1988 k

hence and B are on different z cooridnates and that is wjhy the diagram appears to be wrong

Slope of a plane can be described by describing the slope of the slopes of two lines on the plane, which in this case is very close to 0 for the plane containing A, B and C

As AC + CB = - BA, and this is close to the real numerical values, the contractor has actually made aplae that is vewry close to the real plane ( ideal) if it was made perpendicular to the z axis

if only 2 digits of accurtacy are used, then the error in determining the accuracy of the plane increases

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