Fall, 2014
Lab 4: Volume
Tree Volumes
Reference: T. Avery and H. Burkhart, Forest Measurements, 5th Ed., McGraw-Hill, 2002.
In this lab, we will be estimating the volume of the trunk of a tree. This is roughly a solid of revolution, so the
techniques we have studied in class will be useful. We will give an overview of the problem in Section I,
followed by estimating a volume integral with a sequence of diameter measurements and the trapezoidal rule in
Section II. In Section III, we see how to create a model using a single statistic for the tree (called diameter at
breast height) to estimate the diameter of the tree at different heights and thus the volume.
Goals for this Lab
The goals for lab 4 are as follows:
• To find the volume of a solid of revolution using numerical methods;
• To learn how a model for tree diameters might be developed, and to use such a model to predict the
volume of a tree from simple measurements.
Honor Code Policies
Please read and understand the following before you begin. Honor code policies established by your teacher
will govern this lab.
You may complete the lab in a small group of 2-3 people. If you choose to complete the lab in a group, all
members must be present (at the same place at the same time) during the completion and write-up of all answers
and explanations to all problems in the lab. Each person involved must contribute to the work. Each person
should in fact spend some time working directly with Excel; it cannot be left to one person to do all of the Excel
work in the lab. Each person in the group should be at the keyboard typing for at least one of the problems,
other members of the group should still help and be involved even when they are not sitting at the computer.
Each person who contributed to the lab should get a copy of the completed work to use in answering quiz
questions. (We recommend for this reason that you create a complete, neat, and well-organized report, but this
is up to you.) No one who did not contribute to the lab should get a copy of this report.
Finally, each person will take the quiz individually. When you take the quiz, your work should be your own
(based on the lab work you did previously). You are not permitted to discuss the questions or answers with
anyone other than your instructor under any circumstances. To discuss the questions or answers with anyone
else will be considered an Honor Code violation.
Part I: Introductory Material
This lab concerns the calculation of the volume of a tree trunk. We will assume that the cross section of the tree
at any height is circular. In this lab, h will denote the height, in feet, of some point on the trunk, and d will
denote the diameter in inches at height h. The value of d when h = 4.5 is called the diameter at breast height,
and denoted B. This is a convenient standard measurement of the tree’s size. For various tree species, other
dimensions (such as total height, denoted H) can be estimated from B. The calculations in this lab will give you
a rough idea how such tables for volume can be constructed.
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Portion of Tree Trunk with Constant Diameter
A slice (segment) of the trunk
The volume of a circular cylinder is given by V = π r Δh
where r is the radius and ∆h is the thickness. This could also be written as
2
V =π
∫
b
a
r 2 dh = π r 2 ∫ a dh = π r2 Δh (since r 2 is a constant)
b
(Here, ∆h = (b – a).) Note that the last step is only valid if the radius is constant. Otherwise, we must evaluate
the integral in some other way. Also, the units of r and h should be the same, so that the answer will come out in
consistent units.
A tree will not have constant radius, but its volume could be estimated numerically using small slices of almost
constant radius, as we have done in class. Furthermore, if we could come up with a function r(h) for the radius
at height h, we could calculate (or at least estimate) the required integral from height h = a to h = b to find the
volume. We will develop a model for d 2 in Part III of this lab, but we will have to convert d 2 to r 2 and adjust
the units.
Now complete the “Are You Ready” quiz and check your answers in the on-line help for Lab 4
"Are You Ready" Quiz
Instructions: Read Part I: Introductory Material before attempting this quiz. Then take the quiz and check your
answers at the on-line help site for Lab 4. The answers will be useful to you throughout the rest of the lab.
1. The volume of a circular cylinder (constant radius r and height Δh ) is given by
V=
2. The volume of a vertical object with circular cross sections of varying radius r( h) , from height h = a to
height h = b is given by
V=
(If you need a reminder, see your notes from Lesson 16.)
3. The breast height for trees is defined as h = ______ feet. The value for diameter, d, at this height is
called ____________________________________________________, and labeled B. The total height
of the tree is denoted _____.
4. Suppose the diameter of a circle is 6 inches. Convert this value to the radius of the circle in feet:

6
??
(d in inches)
This answer for radius in feet is found by the following:

(r in feet)
=
.
What constant should be filled in for the missing value "??" in the denominator of the above fraction?
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Fall, 2014
Put this constant into the first fraction below. Then complete the simplification to get the second fraction
written below.
⎛
r2 = ⎜
⎝
⎞2
⎟ =
⎠
d
d2
You have now figured out how to convert d 2 in inches into r 2 in feet. You will need to use this conversion
factor in the lab.
Consider this information from the Scholar website in Spring 2012: By using only online evaluations, the
University is reducing printing costs as well as saving trees. There were 86,297 online evaluations fall semester,
2011. With some ballpark calculations from Conservatree, we can estimate that by changing to online
evaluations that semester, the University saved 10 trees!
Calculations: 1 tree makes 16.67 reams of copy paper or 8,333.3 sheets (estimate from Conservatree) Estimated size of tree: tree 40 feet tall, 6-­‐8 inches in diameter (Conservatree) 86297 sheets of paper 86297/8333.3= 10.356 trees Part II: Estimating Volume from Data
We will consider data of the same type of tree considered above. Our tree is 40 feet high (H) and measures 8
inches in diameter at breast height (B). Suppose further that we have measured the diameter of the tree in
inches at various heights above the ground, and obtained the following data:
height (ft)
diameter (in)
0
8.3
5
7.9
10
7.4
15
7
20
6.5
25
6
30
5.6
35
5.3
40
5
For your convenience, this data is provided in an Excel worksheet available under Lab 4 on the Math 2015 web
site, so you can copy and paste any data you need.
1. We will estimate the volume of this tree’s trunk between h = 0 feet and h = 35 feet.
a) What is the integral, using r(h) as the radius of the tree at height h, that will give the volume of the trunk
between h = 0 feet and h = 35 feet in cubic feet?
b) Use the trapezoidal rule with ∆h = 5 to estimate the integral from part a) on a spreadsheet in Excel. To
do this, use the Lab4DataF14 file. Then create a third column for the integrand from part a), using an Excel
formula applied to the data in your table. You should recall that in Excel, π can be entered as PI().
(Warning: Check to make sure your units match on r and h! You may need to convert some units to make
sure your answer is in cubic feet.)
Set up and complete the trapezoidal rule.
(See the Supplement on Numerical Integration after the notes for Lesson 3 or Lab 2 if you need help setting
up the trapezoidal rule in Excel.)
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Fall, 2014
Part III: Estimating Diameters from B
Developing a table as above for all the trees in a forest would be impractical, so various formulas, called taper
models, have been developed for estimating the diameter at different heights from B. (Essentially, the model
describes how diameters taper off as you move up the tree.) One such model, due to Kozak, Munro, and Smith
(cited in Forest Measurements, p. 180), is the following:
d2
h
y = a0 + a1 x + a2 x 2 , where y = 2 and x = .
B
H
(Recall that H is the total height of the tree, and B is diameter at breast height. The variables d and h represent
the diameter d at height h.) Our task will be to figure out what the constants a0, a1, and a2 are. The method is
known as regression, which means fitting a curve (like the above quadratic equation) to data by choosing
appropriate constants. Luckily, Excel can do this for us with what it calls a trendline.
2. Create a taper model of the above type using the data given for the tree in Part II by doing the following.
(You will need to know H and B for this particular tree; recall the definition of these from the “Are You
Ready” quiz; the values for our tree are found in the beginning of Part II.)
a) Create four columns in an Excel worksheet to get the x and y values specified in the model:
i) In the first two columns, enter the height and diameter values from the table in Section II. You do
not need to convert the units here; just enter the data as it is. (Use all of the data available to you
in the table—even data you may not have used in problem 1! You are attempting to generate a
model for the radius at height h, so you want as much data from the tree as possible.)
h
ii) Let the third column be x-values. Recall that in this model, x = , so this is what you must enter in
H
this column. You should enter this as a formula using the first column. (The value of H is given
before the table of values for the tree.)
d2
iii) The fourth column should be y-values, which should be y = 2 . (Leave d and B in inches. Use a
B
formula in Excel and the d values from column 2.)
b) Make a spreadsheet chart (graph) of x vs. y for the data in part a). Note: You need to select only the
data from the x and y columns of the table you created in part a), as you are looking for a relationship
h
d2
between
and 2 .
H
B
i) Choose an xy-scatter plot, as we do for all graphs in this class, but this time you must select the first
sub-plot type: just points. There should be no curves connecting the markers.
ii) Complete the graph, titling it “Taper Model” and eliminating the legend, which will not be needed.
c) After you have completed your chart, you are ready to insert a trendline.
i) Click on one data point. All the data should become highlighted. Now choose “Add Trendline…”
under the Chart menu (at the top of the screen).
ii) You must now select the correct type of trendline. Choose Type from the left of the dialogue box
that opens. Our model is quadratic, so we must choose a polynomial of degree two. (But don’t click
OK yet.)
iii) We must also make sure that the equation appears on the chart. To do this, click on Options from
the list in the same box and select “Display equation on chart.” Then click “OK.” (You should not
select any other options.)
You should now have a second-degree polynomial graphed with your data, and the equation
displayed. Notice how the trendline does not exactly go through the points, but comes close to most
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Fall, 2014
of them. This is the quadratic polynomial that comes “closest” to all of the points plotted, so it will
be our model for the tree.
(For more help inserting a trendline, see the online Excel help for this class, or Excel’s built in help.)
d) Rewrite the formula (on paper) that you obtained in Excel for the trendline, appropriately substituting in
h
d2
for y and x. In this formula, leave the total height and the diameter at breast height as just
2 and
H
B
the letters B and H, rather than plugging in the specific numbers for this tree. (This will allow you to use
the formula to estimate d 2 for different trees.) Solve for d 2 as a function of h.
h
h2
When you are all done, your answer should have the form d 2 = B 2 (a0 + a1 + a2 2 ),
H
H
where the a0 , a1 and a2 are numbers from your regression formula.
You now have a model which gives the predicted diameter squared at a given height h, based only on total
height H and the diameter at breast height B.
3. a) To check the accuracy of your regression formula, use your taper model to predict the diameter (in
inches) of the tree in Part I at height h = 20 feet. (Note that your model gives d 2 , and not d!)
How well does your model predict the diameter at 20 feet, according to our data?
b) If we assume that our model is valid for other trees of the same type, what is the approximate diameter at
h = 20 feet for a tree with height, H = 55 feet and diameter at breast height, B = 10 inches?
4. We will use our integral formula for volume and the regression formula for d 2 that you found in Question 2
to make a new estimate for the volume of our tree between 0 and 35 feet above the ground. (Use the B and
H that were already given for the tree at the beginning of Part II.)
a) Start by writing out the correct integral. What is the integral using d 2 in inches for the volume of the tree
between h = 0 and h = 35 feet? (Hint: See your “Are You Ready” quiz to determine how to convert d 2 in
inches into r 2 in feet.) Next, fill in your regression formula for d 2 and appropriate values for B, H, and the
limits on the integral. When you are done, you should have an integral that is ready to be evaluated.
b) Evaluate the integral you found in part a). There are several possible ways to do this, please use the
integration rules we learned in class for this type of equation and show your work. It is strongly
recommended that you use Excel for your calculations.
5. Compare your answer in 4b) to the answer you found using the trapezoidal rule in problem 1. (It should be
similar!)
Is the value of the integral in 4a) the actual volume of the wood between 0 and 35 feet? Or would that just
be another estimate? (Note: It is possible to exactly evaluate the integral in 4a), so do not claim that it is an
estimate simply because you chose to estimate the integral in 4b) rather than calculating it exactly. Also
ignore errors that are only due to rounding.)
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Fall, 2014
If you feel that the exact value of the integral would give the exact volume of the tree, explain why.
If you feel that this is still just an estimate, explain exactly in what ways the answer is an estimate, and not
the exact volume.
6. Let’s go back to our second tree from problem 3b), with height 55 feet and diameter at breast height of 10
inches. Assume that the regression formula you developed in Question 2 is still valid (but with the new H
and B values).
a) Write down a new integral for the volume of this tree between h = 0 and h = 35 feet.
b) Evaluate the integral from part a) as you did in (4). Show your work.
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