The Tower of Pisa in Italy has its famous lean to the south because the clay
and sand ground on which it is built is softer on the south side than the
north. The tilt is often found by measuring the distance that the upper part
of the tower overhangs the base, indicated by h in the figure below. In 1980 ,
the tower had a tilt of \(\mathrm{h}=4.49 \mathrm{~m}\), and this tilt was
increasing by about \(1 \mathrm{~mm} /\) year.
We will investigate how the tilt of the tower changed from 1980 to \(1995 .\)
a) First, note that our units do not match: The tilt in 1980 was given as
\(4.49 \mathrm{~m}\), but the annual increase in the tilt is given as \(1
\mathrm{~mm} /\) year. Our first goal is to make the units the same. We will
use millimeters \((\mathrm{mm})\). Convert \(4.49 \mathrm{~m}\) to \(\mathrm{mm}\).
b) Get a sheet of graph paper. Since the tilt of the tower depends on the
year, make the year the independent variable and place it on the horizontal
axis. Let t represent the year. Make the tilt the dependent variable and place
it on the vertical axis. Let h represent the tilt, measured in millimeters
(mm). Choose 1980 as the first year on the horizontal axis and mark every year
thereafter, until 1995 . Let the vertical axis begin at \(4.49 \mathrm{~m}\),
converted to \(\mathrm{mm}\) from part (a), since that was our first
measurement; and then we mark every \(1 \mathrm{~mm}\) thereafter up to \(4510
\mathrm{~mm}\).
c) Think of 1980 as the starting year. Together with the tilt measurement from
that year, it forms a point. What are the coordinates of this point? Plot the
point on your coordinate system.
d) Beginning at the first point, from part (c), move one year to the right (to
1981 ) and \(1 \mathrm{~mm}\) up (because the tilt increases) and plot a new
data point.
e) Each time you move one year to the right, you must move \(1 \mathrm{~mm}\) up
and plot a new point. Repeat this process until you reach the year \(1995 .\)
f) Keeping in mind that we are modeling this discrete situation continuously,
draw a line through your data points. We can use this model to make
predictions.
g) According to computer simulation models, which use sophisticated
mathematics, the tower would be in danger of collapsing when h reaches about
\(4495 \mathrm{~mm}\). Use your graph to estimate what year this would happen.
h) In reality, the tilt of the tower passed \(4495 \mathrm{~mm}\) and the tower
did not collapse. In fact, the tilt increased to \(4500 \mathrm{~mm}\) before
the tower was closed on January 7,1990 , to undergo renovations to decrease
the tilt. (The tower was reopened in 2001, after engineers used weights and
removed dirt from under the base to decrease the tilt by \(450 \mathrm{~mm}\).)
What might be some reasons why the prediction of the computer model was wrong?
i) The following table lists the tilt of the tower, h, the year, and the
number of years since 1980 . In 1980 , the tilt was \(4490 \mathrm{~mm}\) and no
occurrences of the \(1 \mathrm{~mm}\) increase had happened yet, so we fill in
\(4490+0(1)=4490\). In 1981 , one occurrence of the \(1 \mathrm{~mm}\) increase
had occurred because one year had passed since 1980 . Therefore, the tilt was
\(4490+1\) (1). In 1982 , two occurrences of the \(1 \mathrm{~mm}\) increase had
occurred, because 2 years had passed since 1980 . Thus, the tilt was
\(4490+2(1)\). And the pattern continues in this manner. Fill in the remaining
entries.
j) Let \(x\) represent the number of years since 1980 and \(\mathrm{h}\) represent
the tilt. Using the table above, write an equation that relates \(\mathrm{h}\)
and \(\mathrm{x}\).
k) Use your equation to predict the tilt in 1990 . Does it agree with the
actual value from 1990 ? Does it agree with the value that is shown on the
graph you made?
l) In part (g), you used the graph to predict the year in which the tilt would
be 4495mm. Use your equation to make the same prediction. Do the answers
agree?
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine 
