Calculus 112 Winter 2012 Project
Submitted by: Kendra MacInnis
Submitted to: Dr. Taylor
Submitted by: Michelle Aucoin
Submitted to: Dr. Lukeman
April 2
Introduction
A danger to marine travel has been, for a long time, icebergs in the Atlantic Ocean. For
any given iceberg, the portion that sits below the water surface is the majority of its volume.
Using calculus, icebergs and the danger they give marine travel will be measured. We will first
understand iceberg volumes. The result of this project will be to understand the depth of icebergs
in relation to what is able to be seen above water. First, the percentage of the volume of the
iceberg above the surface of the water has to be proven. Then, with this formula and the densities
of water and ice, the volume of an iceberg above and below the surface of the water is calculated.
Second, a formula is determined for the volume of a segment of a spherical iceberg of radius r
and height h and another formula for the volume of a segment of a conical iceberg of height h, a
base radius r and a total height of 3r. In the third and final step, the underwater depth for a
spherical and conical iceberg will be determined using a radius of 10 m.
Problem Statement and Approach
In step one, the percentage of the volume of the object (iceberg) above the surface of the
liquid (water) was proven as
This was proven by using the buoyant force given
by F=
These formula
and the weight of the object given by
.
were used because Archimedes' Principle states that the buoyancy force on an object partially or
fully submerged in a fluid is equal to the weight of the fluid the object displaces. Therefore by
setting these two formula equal to each other,
could be solved for. An expression that
represents the percentage of iceberg above the water was found by dividing the volume of
iceberg above the water by the total volume of the iceberg. Another way of doing this is to take
the total volume of the iceberg and subtract from it the volume of the iceberg below the water all
divided by the total volume of the iceberg. Into this simplified expression,
was
plugged in. Simplified, it was proven that the percentage of the volume of the iceberg above the
water surface is given by
With this, the percentage of the volume of an iceberg
above the surface and below the surface can be determined by using the densities of ice and
seawater.
In step two, a formula for the volume of a segment of a spherical iceberg of height h and
a formula for the volume of a segment of a conical iceberg of height h, base radius r, and total
height of 3r are determined. For a spherical iceberg, by the Pythagorean Theorem
,
where y is (r-y), x is equal to
which is the radius of the segment. Therefore, by
integration, the volume of a segment of the spherical iceberg of height h is determined. For the
conical iceberg, by similar triangles, x was again the radius of the segment which was found
equal to
. By integration, the volume of a segment of the conical iceberg is found.
In step three, the underwater depth for a spherical and conical iceberg are determined
using a radius of 10m. For a spherical iceberg, the volume is given by V=
. This is
multiplied by the percentage volume of an iceberg below water, which gives
. Then
this will be set equal to the volume of the segment found in step two. Solving for h, the height of
the iceberg below the water is found. By a simple calculation of the total height (2r) minus the
height below the water, the height of the iceberg above the water is found. The same process is
repeated for the conical iceberg, only the volume is given by V=
where r =10 and h=30.
Results and Conclusion
In conclusion, the percentage of the volume above water of any given iceberg and a
formula for the volume of a segment of spherical and conical icebergs were used to determine
the depth of an iceberg below the water surface. Since only 10% of an iceberg is visible above
water it is difficult to visually establish the volume of the iceberg. Therefore, the information
calculated in this project would be useful in marine travel, for example, ships could use this to
ensure they do not hit an iceberg. The shape of icebergs vary and in this project spherical and
conical icebergs were dealt with. The underwater depths for these icebergs were calculated. The
underwater depth of a spherical iceberg of radius 10 meters was found to be approximately 15.98
meters. The underwater depth of a conical iceberg of base radius 10 meters, and total height 30
meters was found to be approximately 15.86 meters. Therefore, the spherical iceberg is deeper in
the water. However, the spherical iceberg only has a visible height of approximately 4.016
meters, whereas the conical iceberg has a much greater visible height of 14.14 meters. Therefore,
the conical shaped iceberg is higher above water.