Math 116 — Second Midterm
November 18, 2009
Name:
EXAM SOLUTIONS
Instructor:
Section:
1.
Do not open this exam until you are told to do so.
2.
This exam has 9 pages including this cover. There are 8 problems. Note that the problems
are not of equal difficulty, so you may want to skip over and return to a problem on which
you are stuck.
3.
Do not separate the pages of this exam. If they do become separated, write your name on
every page and point this out to your instructor when you hand in the exam.
4.
Please read the instructions for each individual problem carefully. One of the skills being
tested on this exam is your ability to interpret mathematical questions, so instructors will
not answer questions about exam problems during the exam.
5.
Show an appropriate amount of work (including appropriate explanation) for each problem,
so that graders can see not only your answer but how you obtained it. Include units in your
answer where that is appropriate.
6.
You may use any calculator except a TI92 (or other calculator with a full alphanumeric
keypad). However, you must show work for any calculation which we have learned how to
do in this course. You are also allowed two sides of a 3
00
×
5
00
note card.
7.
If you use graphs or tables to find an answer, be sure to include an explanation and sketch
of the graph, and to write out the entries of the table that you use.
8.
Turn off all cell phones and pagers
, and remove all headphones.
Problem
Points
Score
1
12
2
12
3
12
4
14
5
12
6
9
7
16
8
13
Total
100
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Math 116 / Exam 2 (November 18, 2009)
page 2
1
. [12 points] A window is being rated for underwater pressure tests. The window surface is the
region bounded by the
x
axis, the line
x
= 4 and the graph of
y
=
x/
2, where all lengths are
measured in feet. (Note that the density of water is 62.4 pounds per cubic foot.)
a
. [4 points]
Suppose the window is lowered under water so that the top of the window is
F
feet below the surface. Write an expression that estimates the force due to pressure on
a thin horizontal strip of the window
h
i
feet below the top of the window, given that the
width of the strip is Δ
h
.
Solution:
The force due to hydrostatic pressure is approximately given by the product
of the density of water, the depth of the slice, and the area of the slice. A slice of width
Δ
h
that is
h
i
feet below the top of the window has depth
F
+
h
i
. The pressure along
this strip is 62
.
4(
F
+
h
i
) pounds per square foot. The width of the strip of window
h
i
feet below the top point is given by
w
i
= 2
h
i
, and so the area of the strip is 2
h
i
Δ
h
. The
force due to pressure along the strip is then given by
Force
=
Pressure
×
Area
= 62
.
4(
F
+
h
i
)(2
h
i
Δ
h
) = 124
.
8
h
i
(
F
+
h
i
)Δ
h
b
. [5 points] Set up a definite integral that represents the total force due to pressure on the
window if the top of the window is
F
feet under water. Evaluate this integral in order
to determine the total force due to pressure on the window. Show appropriate work to
support your answer, and include units in your final answer. Your answer may be in terms
of
F
.
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