Lab 14 Soybean Volume Assignment
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Michigan Technological University *
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Course
3160
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
5
Uploaded by MateCamel1550
Lab 14: Soybean Volume
◼
Instructions
Welcome to the assignment for the Soybean Volume Lab. The purpose of this assignment is to find the
volume of the soybeans in the silo described in the Lab 14 Soybean Volume introduction notebook:
1. Save the notebook to your computer right away, and save your work often.
2. When you have finished all the exercises, close any unnecessary cells and save as a PDF. (Make sure
you
Rasterize
all 3-dimensional plots)
3. Upload the PDF on Canvas.
4.
Do not delete this notebook.
Save all your work until after the semester is over and your final grade has
been issued.
Due Date:
see Canvas
◼
Volume in the second quadrant
We already defined points
a
,
b
, and
c
as
(
0, 0,
ha
)
,
(
10, 0,
hb
)
, and
(
0, 10,
hc
)
, respectively in the introduc-
tion notebook. We now define points
d
and
e
similarly below (go ahead and evaluate this cell):
In[1]:=
a
= {
0, 0, ha
}
;
b
= {
10, 0, hb
}
;
c
= {
0, 10, hc
}
;
d
= {-
10, 0, hd
}
;
e
= {
0,
-
10, he
}
;
Question 1
: Follow the procedure outlined in the Soybean Volume Lab notebook to find the equation of the
plane above the second quadrant (passing through points
a
,
c
and
d
). Give the equation in rectangular
coordinates and write your answer in an input cell in the form “
z
=
. . . "
In[6]:=
u
=
d
-
a;
v
=
c
-
a;
Cross
[
u, v
]
Out[8]=
{
10 ha
-
10 hd,
-
10 ha
+
10 hc,
-
100
}
In[9]:=
Solve
[(
10 ha
-
10 hd
) (
x
-
0
) + (-
10 ha
+
10 hc
) (
y
-
0
) -
100
* (
z
-
ha
)
0, z
]
Out[9]=
z
1
10
×
(
10 ha
+
ha x
-
hd x
-
ha y
+
hc y
)
Question 2
: Convert the equation of the plane in
Question 1
into cylindrical coordinates and define the
function p2
(
r
,
θ
)
to be the equation of this plane.
Printed by Wolfram Mathematica Student Edition
In[10]:=
1
10
×
(
10 ha
+
ha x
-
hd x
-
ha y
+
hc y
) /
. x
r
*
Cos
[
θ
] /
. y
r
*
Sin
[
θ
]
Out[10]=
1
10
×
(
10 ha
+
ha r Cos
[
θ
] -
hd r Cos
[
θ
] -
ha r Sin
[
θ
] +
hc r Sin
[
θ
])
In[11]:=
p2
[
r
_
,
θ
_] =
1
10
×
(
10 ha
+
ha r Cos
[
θ
] -
hd r Cos
[
θ
] -
ha r Sin
[
θ
] +
hc r Sin
[
θ
])
Out[11]=
1
10
×
(
10 ha
+
ha r Cos
[
θ
] -
hd r Cos
[
θ
] -
ha r Sin
[
θ
] +
hc r Sin
[
θ
])
Question 3
: To find the volume of beans in the second quadrant, we will again use a double integral in
cylindrical coordinates. Write down the upper and lower limits of integration for the variable
θ
in the integral:
Lower limit:
θ
=
π
2
Upper limit:
θ
=
π
Question 4
: In the cell below, write the double integral that gives the volume of soybeans in the second
quadrant.
In[12]:=
π
2
π
0
10
p2
[
r,
θ
]
r
r
θ
Out[12]=
25
3
×
(
4
(
hc
+
hd
) +
ha
(-
8
+
3
π
))
Question 5
: What you calculated in question 4 is the volume of the soybeans in the second quadrant in
terms of the heights
ha
,
hc
and
hd
. Plug in sample values of
ha
=
hc
=
hd
=
30 into this volume expression.
The shape corresponding to these values is a quarter of a cylinder with radius 10 and height 30. Does your
answer for the volume correspond to the correct value of 750
π
?
(If not, you should carefully work through
questions 1-4 to see what went wrong.)
In[13]:=
ha
=
hb
=
hc
=
hd
=
30;
25
3
×
(
4
(
hc
+
hd
) +
ha
(-
8
+
3
π
))
Out[14]=
25
3
×
(
240
+
30
×
(-
8
+
3
π
))
In[15]:=
π
Rationalize
N
25
3
×
(
240
+
30
×
(-
8
+
3
π
))
π
Out[15]=
750
π
2
Lab 14 Soybean Volume Assignment.nb
Printed by Wolfram Mathematica Student Edition
◼
Volume in quadrants 3 and 4
Question 6
: Following the same steps as in the previous section, write the expression (in terms of
ha
,
hd
and
he
) that gives the volume of the beans in the third quadrant.
In[16]:=
Solve
[
Dot
[
Cross
[
d
-
a, e
-
a
]
,
({
x, y, z
} -
a
)]
0, z
]
Out[16]=
z
1
10
×
(
300
+
30 y
-
he y
)
In[17]:=
1
10
×
(
10 ha
+
ha x
-
hd x
+
ha y
-
he y
) /
. x
r
*
Cos
[
θ
] /
. y
r
*
Sin
[
θ
]
Out[17]=
1
10
×
(
300
+
30 r Sin
[
θ
] -
he r Sin
[
θ
])
In[21]:=
p3
[
r
_
,
θ
_] =
1
10
×
(
300
+
30 r Sin
[
θ
] -
he r Sin
[
θ
])
;
he
=
30;
π
3
π
2
0
10
p3
[
r,
θ
]
r
r
θ
Out[23]=
750
π
Question 7
: Write the expression (in terms of
ha
,
hb
and
he
) that gives the volume of the soybeans in the
fourth quadrant.
In[24]:=
Solve
[
Dot
[
Cross
[
e
-
a, b
-
a
]
,
({
x, y, z
} -
a
)]
0, z
]
Out[24]=
{{
z
30
}}
In[25]:=
p4
[
r
_
,
θ
_] =
30;
3
π
2
2
π
0
10
p4
[
r,
θ
]
r
r
θ
Out[26]=
750
π
◼
Putting it all together
Question 8
: Define a function in the input cell below called
totalVolume
that takes the list {
ha
,
hb
,
hc
,
hd,
he
} as input and returns the sum of the volume formulas from all four quadrants (Note: It is important that
you define
totalVolume
so that it takes a
list
of five elements as input). The volume from the first quadrant
can be copied from the Soybean Lab introduction notebook.
Lab 14 Soybean Volume Assignment.nb
3
Printed by Wolfram Mathematica Student Edition
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