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Engineering Activity
Scaling height of heights!
Introduction
Some objects and distances in our world are very difficult—even impossible—
to directly measure by hand or with tools. Engineers who design different
types of structures, either incredibly large or very small, or machines, which
might travel distances deep under the sea or far into space, must have an
accurate understanding of lengths and dimensions, even when they cannot be
measured. Often in these cases, engineers use trigonometry and other
mathematical relationships to find very close approximations to lengths and
dimensions. In the design of large structures, engineers must ensure that
forces acting on the structure will balance so that they remain stationary.
Engineers use trigonometry to account for vertical and horizontal components
of the different forces that act on structures, thus determining that the
structure will be able to stand without collapsing before it is even constructed.
These strategies, based on mathematical knowledge, enable engineers to
design solutions to problems that might otherwise be unsolvable.
The Basics of Trigonometry
Trigonometric Ratios
TRIGONOMETRIC VALUES:
Terms used in Heights and Distance
Below we can see terms used in the applications of trigonometry:
Horizontal Ray:
A ray parallel to the surface of the earth emerging from the eye of the
observer is known as horizontal ray.
Ray of Vision:
The ray from the eye of the observer towards the object under observation is
known as the ray of vision or ray of sight.
Angle of Elevation:
If the object under observation is above the horizontal ray passing through the
point of observation, the measure of the angle formed by the horizontal ray
and the ray of vision is known as angle of elevation.
Angle of Depression:
If the object under observation is below the horizontal ray passing through the
point of observation, the measure of the angle formed by the horizontal ray
and the ray of vision is known as angle of depression.
Applications of Right Triangle Trigonometry
Given below are some of the real world applications of trigonometry. Trigonometry is
the branch of mathematics that studies triangles and their relationships. Trigonometry is
commonly used in finding the height of towers, mountains and also used to find the
distance of the shore from a point in the sea etc. Trigonometry provides perspective on
real world events. It is used in satellite systems and astronomy, architecture,
engineering, geography and many other fields.
Constructing a Clinometer
Create a clinometer, by following these steps:
a.
b.
c.
d.
Cut out the photocopied protractor.
Glue it to your piece of cardboard.
Cut out the cardboard so that it is the shape of your protractor
Cut a 4-inch piece of string; fasten it to the middle of the protractor by
tying it and using tape. It is important that you fasten your string before
you tape your straw.
e. Tape your straw to the top flat end of your protractor.
f. At the end of the string, fasten a couple of washers to it by tying a knot.
g. Your clinometer should look like this:
Using a Clinometer to measure height of a tree
Evaluation:
ALTERNATE METHODS OF MEASURING INACCESSIBLE HEIGHTS
Method 1:
One unusual method, which may not be as accurate, is to drop a ball from the
top and monitor the time taken to reach the ground. This works well if the
building is tall and straight. If the building is tapered or larger at the base the
method will still deliver an approximate height of the building.
You are probably familiar with this simple physics formula:
H = 0.5 * g * t^2,
Where “g” is the constant of gravitational acceleration, and g = 9.8
meters/second2
And “t” is the duration of fall in seconds.
Thus we get “H” which is the distance travelled by the free-falling object,
inferring the height of the building in meters.
Method 2:
This method may be called the “Yardstick” method. It requires
an additional person and a yardstick. It is essential to know the exact height of
your friend, as he/ she becomes the main reference, based on which the
height of the building or a tall tower is calculated.
1. Ask your friend to stand close to the building base or the base of the tower.
2. Hold the yardstick vertically up and move back to match the height of
your friend exactly to an inch in the yardstick.
3. Without moving from that point and holding the yardstick in the same
position, measure between the top and bottom, the number of inches the
tower or the building occupies.
4. Every inch equals the height of your friend. For better understanding, if
your friend is actually 6 feet tall, and the tower or the building makes 50
inches in the yardstick, then the height of the tower or the building is 6 * 50
= 300 feet.