Figure 1
Modeling Problems Inspired by Hurricane Irene
Prepared by:
Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
email:
malkevitch@york.cuny.edu
web page:
http://york.cuny.edu/~malk
Recently, hurricane Irene caused major devastation along the East Coast. Power outages, downed trees, and flooding were common in states such as New Jersey, New York, and Vermont.
Teachers should consider teaching out of the "newspaper" - preparing lessons that support topics that they must cover in their mathematics courses by building on recent events which lead to newspaper stories or have affected their students' lives. The following questions were inspired by a tree that fell down in front of my house when hurricane Irene visited my neighborhood (August 28, 2011).
1. What would be a good mathematical model for the trunk of a tree?
Ideas and comments:
For some kinds of trees, a reasonable model is a cylinder.
2. a. What is a practical method of finding the radius of a tree trunk which is close to a cylinder in shape?
Ideas and comments:
Directly measuring the radius or diameter of a tree trunk is not practical. However, since the circumference C of a circle is related to its radius or diameter, one can use the fact that C = 2ır. This can be done with a tape measure or a piece of string that can be wrapped around the trunk and then measured after it is removed. Have students brainstorm how to do this for getting an answer both in inches and in meters (metric system). If one cuts a cylinder perpendicular to its axis, one gets a circle. If someone saws a cylindrical tree trunk using a "plane" not perpendicular to its axis one gets an ellipse.
2. Find the volume of a tree trunk under the assumption that it is a cylinder?
Ideas and comments:
Figure 1 shows a diagram of a cylinder.
The volume of a cylinder is given by: V = ır2h.
For tree trunk calculations you might work with lengths that yield answers in:
a. cubic inches
b. cubic meters
3. The trunk T of a downed tree that was blocking a street was cut free from its root ball and crown (the top section where multiple branches come out) into a cylinder.
Find a formula for the weight of T if:
a. The tree was an oak.
b. The tree was a white ash.
c. The tree was a maple.
Explain why different sources might give different numbers for the density of wood such as oak.
Ideas and comments:
It may be helpful to review the difference between the concepts of weight and mass. On the Earth's moon you will weigh less than what you weigh on Earth but your mass on the Earth and the Moon will be the same. A mass of 1 kilogram will weigh approximately the same anywhere on Earth but in fact on top of a tall mountain there will be a slight difference in weight.
The age and the amount of moisture in wood will effect its density. Also, there are many kinds of oaks, so it might not be surprising that different types of oak trees might differ in density. A cylindrical section of an oak tree's trunk with its bark removed might give a different density than a board which was kiln dried from the same species of oak. Oaks grown in different quality of soil might also differ in density. Some trees have cores that have become hollowed out. Here I will assume that the whole cylindrical section of the tree is solid. If the hollowed out portion is another cylinder whose radius is again a cylinder, one can compute the volume of a hollowed out tree.
Here are densities (in lbs/ft3) that I located for the three types of trees above:
oak: 37-56
white ash: 40-53
maple: 39-47
Students can find a maximum weight, minimum weight, and "mean" weight using these numbers.
4. Brothers David and Emad have decided that they want to cut a section of a downed maple tree near their home down the block to use as firewood during the winter. They are nervous about carrying a section of the tree which is too heavy. Suppose that they agree not to carry more than 100 lbs. If the tree has circumference 30 inches, find the longest section of tree the boys can carry. If M denotes the largest weight they are willing to carry, find a formula for the longest section h they can carry in terms of the circumference of the tree.
Ideas and comments:
What value should be used for the density of the maple tree in the calculations? One choice might be the mean of the density range given. To be on the safe side one could use the maximum density.
Students can also find a formula for the longest section they can carry in terms of the maximum weight M they are willing to take on and d, the density of the wood, and C the circumference of the tree. Other formulas that can be derived would be in terms of the radius of the cylindrical "log" r or its diameter.
Other calculations would involve what is the weight they wind up carrying if it was specified they wanted to carry a log of length 20 ft (assuming the trunk of the tree is this long).