The Exponential Eiffel Tower
Guided Inquiry
Calculus 3 – Math 2421
Name:
Due Date:
Instructions:
• Carefully read the problem statement and tasks below. You must TYPE your solutions to the tasks
along with all supporting details. This includes restating the problem statement.
• When typing you must
– Explain your solutions fully using proper English.
– If an explanation requires a figure, then sketch the figure on engineering paper and then insert
that page into the report. You may denote it as Figure # 1, etc. It is ok to put more than
one figure on an engineering paper page and then reference the figures from within your typed
explanations.
– If you need to type mathematical symbols then use the typical keyboard symbols for the algebra.
For example, if the function is f (x) = e−cx then type f(x)=e^(-cx) or f(x) = exp(-cx).
• Most students lose points for the following reasons:
The report is not typed, one or more of the sections was skipped, inadequate math was shown on one
or more parts, or appropriate explanations were missing on one or more parts.
Problem Statement:
Completed just one month before the opening of the 1889 Exposition Universelle (World’s Fair) in Paris, the
Eiffel Tower is one of the most recognizable landmarks in the world. It rises 300 meters from a 100-metersquare base to a 10-meter-square observation deck. Surprisingly the project’s chief engineer, Gustave Eiffel,
left no detailed structural analysis that explained the design of the tower. Recent investigations of Eiffel’s
notes and communications at the time the tower was built have led to a plausible model that gives a good
fit to the tower’s shape.
In designing the tower, Eiffel’s primary concern was the effect of wind loading on a free-standing structure
of this size. An analysis of the forces on the tower led to the following principal that we call the Eiffel property.
At any height on the tower (AA in Figure 1), the tangent lines to the tower (correspond to the supporting
forces of the tower) must pass through the center of mass of that part of the tower above that height
(corresponding to the point at which a horizontal wind acts).
It turns out that exponential functions have the Eiffel property very nearly, which means that they give
a very good description of the shape of the Eiffel Tower. Consider the functions y = f (x) = e−cx and
y = g(x) = −e−cx , where c is a positive real number. In this position, the graphs look like the Eiffel Tower
on its side. Now fix two points x = a and x = b on the x-axis and let R be the region between the graphs
on the interval [a, b]. We show that the tangent lines to the curves at x = a intersect at the center of mass
(centroid) of R (see Figure 2).
x
A
a
b
A
Figure 2: Simple 2D image showing the Eiffel property.
Figure 1: Simple 2D image showing the Eiffel property. The top function is f (x) and the bottom function is
g(x).
Tasks:
1. Find equations of the tangent liens to the curves y = f (x) and y = g(x) at x = a. Show that these
tangent lines intersect on the x-axis at the point (a + 1/c, 0).
2. Explain why the center of mass is on the x-axis. Recall that the x-coordinate of the center of mass is
Rb
x [f (x) − g(x)] dx
x = Ra b
[f (x) − g(x)] dx
a
To simplify matters, first assume that a = 0. Then use the fact that f (x) = −g(x) to show that
x=
e−cb (cb + 1) − 1
c(e−cb − 1)
3. Now generalize Step 2 by letting a be any real number with 0 ≤ a < b. Show that the center of mass is
x=
e−bc (cb + 1) − e−ca (ca + 1)
c (e−cb − e−ca )
4. Comparing the results from Steps 1 and 3, the center of mass of R does NOT coincide with the
intersection point of the tangent lines. However let’s investigate what happens if the region R is
allowed to extend indefinitely in the positive x-direction by letting b → +∞. Show that with a = 0,
−bc
1
e (cb + 1) − 1
lim x = lim
= ,
b→∞
b→∞
c(e−cb − 1)
c
and more generally for 0 ≤ a < b
lim x = lim
b→∞
b→∞
e−bc (cb + 1) − e−ca (ca + 1)
1
=a+ .
c (e−cb − e−ca )
c
Thus, if the region extends over the interval [a, ∞), the Eiffel property holds.
5. Explain why the Eiffel property holds for the functions y = ±Ae−cx , where A is any positive real
number.
6. The requirement that we must let b → +∞ to obtain the exact Eiffel property suggests that the model
does not apply to the real Eiffel Tower. Let’s see how much of an assumption we have made. Suppose
the profile of the tower is given by f (x) = Ae−cx , where A and c are to be determined. Assume the
base of the tower is at x = 0 and the top of the tower is at x = 300m. The half-width of the tower
at the base is about 50m, which means that f (0) = 50. The top of the tower has half-width of 5m, so
f (300) = 5. Use these two conditions to show that a reasonable fit to the profile of the tower is
f (x) = 50e−0.0077x .
7. Consider the entire tower, 0 ≤ x ≤ 300 and find where the tangent line to f (x) = 60e−0.0077x at x = 0
intersects the x-axis. Then find the x-coordinate of the center of mass for the full tower. How closely
do these points coincide? The fact that they do not coincide is best explained by the fact that a single
exponential functions does not give an ideal fit to the tower. The model may be modified to include
two exponential functions, which come closer to having the Eiffel property. How could we use TWO
exponential functions to model the real tower?
Is there possible a more real-life (engineering consideration) explanation as to why the points do not
coincide?