data, converting it to a more usable form,
usually in x, y,and zcoordinates. For
example, depending on the desired out-
come, the vertical angle and slope dis-
tances can be converted from polar mea-
surements to show differences in
elevations and horizontal distances. The
horizontal distances and angles can also
be converted from polar measurements
to rectangular coordinates. (For more
about coordinates, see “Geometry and
Trigonometry.”)
How is linear algebraused to
determine the stability of structures?
Structural engineers use linear algebra a
great deal, mainly because there are
numerous equations with many unknowns
associated with the analysis of a structure
in equilibrium. Most of the time, these
equations are linear, even when bending
(material deformation) is involved. Linear algebra can also be used for other structural
concerns, because it deals with the study of vectors, vector spaces, linear transformations,
and systems of linear equations. Of course, linear algebra is not only used to understand
structures; almost every subfield in engineering uses these types of mathematical calcula-
tions. (For more about linear algebra and linear equations, see “Algebra.”)
How is math used to calculate the pressure behind a dam?
There are many engineering considerations and calculations needed when building a
dam, the most important being the water pressure behind the structure. Engineers
know that as the elevation of water behind a dam increases, the height and density of
the water causes higher pressures at the bottom of the dam. Thinking in mathemati-
cal terms, the horizontal force acting on the dam is the integral of the water pressure
over the area of the dam that is in contact with the water. The force exerted by the
water pushes horizontally on the dam face, and this is resisted by the force of static
friction between the dam and the bedrock foundation on which it rests. The water also
tries to rotate the dam about a line running along the base of the dam; the torque
resulting from the weight of the dam acts in the opposite sense.
For example, take the water pressure on a dam, such as the Hoover Dam on the
Colorado River between Arizona and Nevada. Before building that dam, engineers 331
MATH IN ENGINEERING
Geometry and trigonometry are essential mathemat-
ical disciplines that surveyors must understand to
measure property boundaries accurately.
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