Mixing
List of realistic extremum problems
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I am a student who would like to become a teacher, so I am currently following courses in education. One of the things I learned, is that students like authentic, realistic problems. However, much of the extremum problems (in one variable) appear very 'fabricated'. I only know a few 'realistic extremum problems':
- minimizing some cost function
- maximizing some return function
- maximizing the volume of a box.
A realistic example of the third problem can be found here, where the volume of a box is maximized. This box is the kind of box you receive a package in: it is closed, with overlapping flaps.
With respect to multivariable functions, I can think of one extra example: determining a regression line.
However, this list seems rather short. Can anyone give extra examples of real-life situations where we wish to maximize/minimize some sort of function?
(For my class, single variable functions are the most interesting, but I equally like multivariable examples.)
secondary-education calculus student-motivation big-list
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Jasper
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up vote
3
down vote
favorite
I am a student who would like to become a teacher, so I am currently following courses in education. One of the things I learned, is that students like authentic, realistic problems. However, much of the extremum problems (in one variable) appear very 'fabricated'. I only know a few 'realistic extremum problems':
- minimizing some cost function
- maximizing some return function
- maximizing the volume of a box.
A realistic example of the third problem can be found here, where the volume of a box is maximized. This box is the kind of box you receive a package in: it is closed, with overlapping flaps.
With respect to multivariable functions, I can think of one extra example: determining a regression line.
However, this list seems rather short. Can anyone give extra examples of real-life situations where we wish to maximize/minimize some sort of function?
(For my class, single variable functions are the most interesting, but I equally like multivariable examples.)
secondary-education calculus student-motivation big-list
edited 12 hours ago
Jasper
2,402716
asked 20 hours ago
Student
1162
New contributor
Student is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Could you summarize the "realistic box problem" that you link to? That link requires Java in the browser, which many people (including myself) disable for security reasons.
– Rory Daulton
6 hours ago
@RoryDaulton: Here is that problem: "A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. If six congruent cuts (denoted in black) are to be made into the cardboard and five folds (denoted by dotted lines) made to adjoin the cuts so that the resulting piece of cardboard is to be folded to form a closeable rectangular box (see figures below). How should this be done to get a box of largest possible volume?"
– Joseph O'Rourke
6 hours ago
@JosephO'Rourke: Thanks! Is there any way to show the figures with the cuts and folds and final box?
– Rory Daulton
5 hours ago
1 covers a lot of ground: number of shale wells in a section, number of salespeople, hours to keep a store open, etc.
– guest
3 hours ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I am a student who would like to become a teacher, so I am currently following courses in education. One of the things I learned, is that students like authentic, realistic problems. However, much of the extremum problems (in one variable) appear very 'fabricated'. I only know a few 'realistic extremum problems':
- minimizing some cost function
- maximizing some return function
- maximizing the volume of a box.
A realistic example of the third problem can be found here, where the volume of a box is maximized. This box is the kind of box you receive a package in: it is closed, with overlapping flaps.
With respect to multivariable functions, I can think of one extra example: determining a regression line.
However, this list seems rather short. Can anyone give extra examples of real-life situations where we wish to maximize/minimize some sort of function?
(For my class, single variable functions are the most interesting, but I equally like multivariable examples.)
secondary-education calculus student-motivation big-list
edited 12 hours ago
Jasper
2,402716
asked 20 hours ago
Student
1162
New contributor
Student is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I am a student who would like to become a teacher, so I am currently following courses in education. One of the things I learned, is that students like authentic, realistic problems. However, much of the extremum problems (in one variable) appear very 'fabricated'. I only know a few 'realistic extremum problems':
- minimizing some cost function
- maximizing some return function
- maximizing the volume of a box.
A realistic example of the third problem can be found here, where the volume of a box is maximized. This box is the kind of box you receive a package in: it is closed, with overlapping flaps.
With respect to multivariable functions, I can think of one extra example: determining a regression line.
However, this list seems rather short. Can anyone give extra examples of real-life situations where we wish to maximize/minimize some sort of function?
(For my class, single variable functions are the most interesting, but I equally like multivariable examples.)
secondary-education calculus student-motivation big-list
secondary-education calculus student-motivation big-list
edited 12 hours ago
Jasper
2,402716
asked 20 hours ago
Student
1162
New contributor
Student is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 12 hours ago
Jasper
2,402716
asked 20 hours ago
Student
1162
New contributor
Student is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 12 hours ago
Jasper
2,402716
edited 12 hours ago
Jasper
2,402716
edited 12 hours ago
Jasper
2,402716
2,402716
asked 20 hours ago
Student
1162
New contributor
Student is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 20 hours ago
Student
1162
asked 20 hours ago
Student
1162
1162
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New contributor
Student is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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Check out our Code of Conduct.
Could you summarize the "realistic box problem" that you link to? That link requires Java in the browser, which many people (including myself) disable for security reasons.
– Rory Daulton
6 hours ago
@RoryDaulton: Here is that problem: "A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. If six congruent cuts (denoted in black) are to be made into the cardboard and five folds (denoted by dotted lines) made to adjoin the cuts so that the resulting piece of cardboard is to be folded to form a closeable rectangular box (see figures below). How should this be done to get a box of largest possible volume?"
– Joseph O'Rourke
6 hours ago
@JosephO'Rourke: Thanks! Is there any way to show the figures with the cuts and folds and final box?
– Rory Daulton
5 hours ago
1 covers a lot of ground: number of shale wells in a section, number of salespeople, hours to keep a store open, etc.
– guest
3 hours ago
add a comment |Â
Could you summarize the "realistic box problem" that you link to? That link requires Java in the browser, which many people (including myself) disable for security reasons.
– Rory Daulton
6 hours ago
@RoryDaulton: Here is that problem: "A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. If six congruent cuts (denoted in black) are to be made into the cardboard and five folds (denoted by dotted lines) made to adjoin the cuts so that the resulting piece of cardboard is to be folded to form a closeable rectangular box (see figures below). How should this be done to get a box of largest possible volume?"
– Joseph O'Rourke
6 hours ago
@JosephO'Rourke: Thanks! Is there any way to show the figures with the cuts and folds and final box?
– Rory Daulton
5 hours ago
1 covers a lot of ground: number of shale wells in a section, number of salespeople, hours to keep a store open, etc.
– guest
3 hours ago
Could you summarize the "realistic box problem" that you link to? That link requires Java in the browser, which many people (including myself) disable for security reasons.
– Rory Daulton
6 hours ago
Could you summarize the "realistic box problem" that you link to? That link requires Java in the browser, which many people (including myself) disable for security reasons.
– Rory Daulton
6 hours ago
@RoryDaulton: Here is that problem: "A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. If six congruent cuts (denoted in black) are to be made into the cardboard and five folds (denoted by dotted lines) made to adjoin the cuts so that the resulting piece of cardboard is to be folded to form a closeable rectangular box (see figures below). How should this be done to get a box of largest possible volume?"
– Joseph O'Rourke
6 hours ago
@RoryDaulton: Here is that problem: "A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. If six congruent cuts (denoted in black) are to be made into the cardboard and five folds (denoted by dotted lines) made to adjoin the cuts so that the resulting piece of cardboard is to be folded to form a closeable rectangular box (see figures below). How should this be done to get a box of largest possible volume?"
– Joseph O'Rourke
6 hours ago
@JosephO'Rourke: Thanks! Is there any way to show the figures with the cuts and folds and final box?
– Rory Daulton
5 hours ago
@JosephO'Rourke: Thanks! Is there any way to show the figures with the cuts and folds and final box?
– Rory Daulton
5 hours ago
1 covers a lot of ground: number of shale wells in a section, number of salespeople, hours to keep a store open, etc.
– guest
3 hours ago
1 covers a lot of ground: number of shale wells in a section, number of salespeople, hours to keep a store open, etc.
– guest
3 hours ago
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
4
down vote
Here are some optimization problems that were harder than a simple homework problem:
Given the pavillion angle of a diamond, what crown angle produces optimal light return?
What percentage of the lignin should be removed from wood, if one wishes to squeeze the wood into a cheap structure similar to carbon fiber with an optimum strength-to-weight ratio?
When choosing the parameters for a map projection, which parameters produce the least worst-case distortion of the needed area of the map? For example, which two latitudes should have undistorted lengths in an Albers Equal-Area map of the 48 conterminous United States?
Notice that in these examples, more effort went into designing the functions to be optimized, and in choosing the parameter to be varied, than went into performing the calculus.
answered 12 hours ago
Jasper
2,402716
Perhaps one could invent a two-variable function $g(x,y)$ of math-effort $x$ vs. optimization set-up effort $y$ then analyze $g$ subject the to the set of all word problems. If this set was compact then we could optimize $g$. Perhaps the choice of $g$ ought to depend on the course of major study for the student ?
– James S. Cook
1 hour ago
add a comment |Â
up vote
4
down vote
Some example types:
- Minimizing potential energy of any realistic physical system. Examples:
- 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
- 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
- 2D: The surface of a soap film (in equilibrium).
- Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)
- Maximizing a utility function subject to a budget constraint.
- Fermat's principle: Light takes the path of least time.
- Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
- In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
- Space trajectories that minimize fuel use (Mission Design).
- When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $epsilon$,
what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since the formula is exact (no truncation error, only round-off error) if $f$ is a polynomial of degree at most $2$. One could pick $f$ a certain function and value of $x$; even so, the exact solution is hard to obtain.]
answered 2 hours ago
user683
2,9421024
add a comment |Â
up vote
3
down vote
There are many volume-of-a-box questions. I like this one, simpler than
what the OP cites:
Given a rectangle,
cut out squares from the corners so you can fold it up to a box, without a top,
of maximal volume.
The rectangle might be specialized to a square, as below.
See also The Math Forum.
    Â
    Â
(Image from patrickJMT YouTube video.)
In response to @RoryDaulton, here is the box problem to which the
OP @Student points:
answered 18 hours ago
Joseph O'Rourke
14.5k33278
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Here are some optimization problems that were harder than a simple homework problem:
Given the pavillion angle of a diamond, what crown angle produces optimal light return?
What percentage of the lignin should be removed from wood, if one wishes to squeeze the wood into a cheap structure similar to carbon fiber with an optimum strength-to-weight ratio?
When choosing the parameters for a map projection, which parameters produce the least worst-case distortion of the needed area of the map? For example, which two latitudes should have undistorted lengths in an Albers Equal-Area map of the 48 conterminous United States?
Notice that in these examples, more effort went into designing the functions to be optimized, and in choosing the parameter to be varied, than went into performing the calculus.
answered 12 hours ago
Jasper
2,402716
Perhaps one could invent a two-variable function $g(x,y)$ of math-effort $x$ vs. optimization set-up effort $y$ then analyze $g$ subject the to the set of all word problems. If this set was compact then we could optimize $g$. Perhaps the choice of $g$ ought to depend on the course of major study for the student ?
– James S. Cook
1 hour ago
add a comment |Â
up vote
4
down vote
Here are some optimization problems that were harder than a simple homework problem:
Given the pavillion angle of a diamond, what crown angle produces optimal light return?
What percentage of the lignin should be removed from wood, if one wishes to squeeze the wood into a cheap structure similar to carbon fiber with an optimum strength-to-weight ratio?
When choosing the parameters for a map projection, which parameters produce the least worst-case distortion of the needed area of the map? For example, which two latitudes should have undistorted lengths in an Albers Equal-Area map of the 48 conterminous United States?
Notice that in these examples, more effort went into designing the functions to be optimized, and in choosing the parameter to be varied, than went into performing the calculus.
answered 12 hours ago
Jasper
2,402716
Perhaps one could invent a two-variable function $g(x,y)$ of math-effort $x$ vs. optimization set-up effort $y$ then analyze $g$ subject the to the set of all word problems. If this set was compact then we could optimize $g$. Perhaps the choice of $g$ ought to depend on the course of major study for the student ?
– James S. Cook
1 hour ago
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Here are some optimization problems that were harder than a simple homework problem:
Given the pavillion angle of a diamond, what crown angle produces optimal light return?
What percentage of the lignin should be removed from wood, if one wishes to squeeze the wood into a cheap structure similar to carbon fiber with an optimum strength-to-weight ratio?
When choosing the parameters for a map projection, which parameters produce the least worst-case distortion of the needed area of the map? For example, which two latitudes should have undistorted lengths in an Albers Equal-Area map of the 48 conterminous United States?
Notice that in these examples, more effort went into designing the functions to be optimized, and in choosing the parameter to be varied, than went into performing the calculus.
answered 12 hours ago
Jasper
2,402716
Here are some optimization problems that were harder than a simple homework problem:
Given the pavillion angle of a diamond, what crown angle produces optimal light return?
What percentage of the lignin should be removed from wood, if one wishes to squeeze the wood into a cheap structure similar to carbon fiber with an optimum strength-to-weight ratio?
When choosing the parameters for a map projection, which parameters produce the least worst-case distortion of the needed area of the map? For example, which two latitudes should have undistorted lengths in an Albers Equal-Area map of the 48 conterminous United States?
Notice that in these examples, more effort went into designing the functions to be optimized, and in choosing the parameter to be varied, than went into performing the calculus.
answered 12 hours ago
Jasper
2,402716
edited 12 hours ago
answered 12 hours ago
Jasper
2,402716
answered 12 hours ago
Jasper
2,402716
answered 12 hours ago
Jasper
2,402716
2,402716
Perhaps one could invent a two-variable function $g(x,y)$ of math-effort $x$ vs. optimization set-up effort $y$ then analyze $g$ subject the to the set of all word problems. If this set was compact then we could optimize $g$. Perhaps the choice of $g$ ought to depend on the course of major study for the student ?
– James S. Cook
1 hour ago
add a comment |Â
Perhaps one could invent a two-variable function $g(x,y)$ of math-effort $x$ vs. optimization set-up effort $y$ then analyze $g$ subject the to the set of all word problems. If this set was compact then we could optimize $g$. Perhaps the choice of $g$ ought to depend on the course of major study for the student ?
– James S. Cook
1 hour ago
Perhaps one could invent a two-variable function $g(x,y)$ of math-effort $x$ vs. optimization set-up effort $y$ then analyze $g$ subject the to the set of all word problems. If this set was compact then we could optimize $g$. Perhaps the choice of $g$ ought to depend on the course of major study for the student ?
– James S. Cook
1 hour ago
Perhaps one could invent a two-variable function $g(x,y)$ of math-effort $x$ vs. optimization set-up effort $y$ then analyze $g$ subject the to the set of all word problems. If this set was compact then we could optimize $g$. Perhaps the choice of $g$ ought to depend on the course of major study for the student ?
– James S. Cook
1 hour ago
add a comment |Â
up vote
4
down vote
Some example types:
- Minimizing potential energy of any realistic physical system. Examples:
- 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
- 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
- 2D: The surface of a soap film (in equilibrium).
- Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)
- Maximizing a utility function subject to a budget constraint.
- Fermat's principle: Light takes the path of least time.
- Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
- In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
- Space trajectories that minimize fuel use (Mission Design).
- When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $epsilon$,
what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since the formula is exact (no truncation error, only round-off error) if $f$ is a polynomial of degree at most $2$. One could pick $f$ a certain function and value of $x$; even so, the exact solution is hard to obtain.]
answered 2 hours ago
user683
2,9421024
add a comment |Â
up vote
4
down vote
Some example types:
- Minimizing potential energy of any realistic physical system. Examples:
- 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
- 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
- 2D: The surface of a soap film (in equilibrium).
- Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)
- Maximizing a utility function subject to a budget constraint.
- Fermat's principle: Light takes the path of least time.
- Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
- In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
- Space trajectories that minimize fuel use (Mission Design).
- When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $epsilon$,
what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since the formula is exact (no truncation error, only round-off error) if $f$ is a polynomial of degree at most $2$. One could pick $f$ a certain function and value of $x$; even so, the exact solution is hard to obtain.]
answered 2 hours ago
user683
2,9421024
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Some example types:
- Minimizing potential energy of any realistic physical system. Examples:
- 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
- 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
- 2D: The surface of a soap film (in equilibrium).
- Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)
- Maximizing a utility function subject to a budget constraint.
- Fermat's principle: Light takes the path of least time.
- Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
- In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
- Space trajectories that minimize fuel use (Mission Design).
- When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $epsilon$,
what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since the formula is exact (no truncation error, only round-off error) if $f$ is a polynomial of degree at most $2$. One could pick $f$ a certain function and value of $x$; even so, the exact solution is hard to obtain.]
answered 2 hours ago
user683
2,9421024
Some example types:
- Minimizing potential energy of any realistic physical system. Examples:
- 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
- 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
- 2D: The surface of a soap film (in equilibrium).
- Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)
- Maximizing a utility function subject to a budget constraint.
- Fermat's principle: Light takes the path of least time.
- Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
- In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
- Space trajectories that minimize fuel use (Mission Design).
- When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $epsilon$,
what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since the formula is exact (no truncation error, only round-off error) if $f$ is a polynomial of degree at most $2$. One could pick $f$ a certain function and value of $x$; even so, the exact solution is hard to obtain.]
answered 2 hours ago
user683
2,9421024
edited 32 mins ago
answered 2 hours ago
user683
2,9421024
answered 2 hours ago
user683
2,9421024
answered 2 hours ago
user683
2,9421024
2,9421024
add a comment |Â
add a comment |Â
up vote
3
down vote
There are many volume-of-a-box questions. I like this one, simpler than
what the OP cites:
Given a rectangle,
cut out squares from the corners so you can fold it up to a box, without a top,
of maximal volume.
The rectangle might be specialized to a square, as below.
See also The Math Forum.
    Â
    Â
(Image from patrickJMT YouTube video.)
In response to @RoryDaulton, here is the box problem to which the
OP @Student points:
answered 18 hours ago
Joseph O'Rourke
14.5k33278
add a comment |Â
up vote
3
down vote
There are many volume-of-a-box questions. I like this one, simpler than
what the OP cites:
Given a rectangle,
cut out squares from the corners so you can fold it up to a box, without a top,
of maximal volume.
The rectangle might be specialized to a square, as below.
See also The Math Forum.
    Â
    Â
(Image from patrickJMT YouTube video.)
In response to @RoryDaulton, here is the box problem to which the
OP @Student points:
answered 18 hours ago
Joseph O'Rourke
14.5k33278
add a comment |Â
up vote
3
down vote
up vote
3
down vote
There are many volume-of-a-box questions. I like this one, simpler than
what the OP cites:
Given a rectangle,
cut out squares from the corners so you can fold it up to a box, without a top,
of maximal volume.
The rectangle might be specialized to a square, as below.
See also The Math Forum.
    Â
    Â
(Image from patrickJMT YouTube video.)
In response to @RoryDaulton, here is the box problem to which the
OP @Student points:
answered 18 hours ago
Joseph O'Rourke
14.5k33278
There are many volume-of-a-box questions. I like this one, simpler than
what the OP cites:
Given a rectangle,
cut out squares from the corners so you can fold it up to a box, without a top,
of maximal volume.
The rectangle might be specialized to a square, as below.
See also The Math Forum.
    Â
    Â
(Image from patrickJMT YouTube video.)
In response to @RoryDaulton, here is the box problem to which the
OP @Student points:
answered 18 hours ago
Joseph O'Rourke
14.5k33278
edited 4 hours ago
answered 18 hours ago
Joseph O'Rourke
14.5k33278
answered 18 hours ago
Joseph O'Rourke
14.5k33278
answered 18 hours ago
Joseph O'Rourke
14.5k33278
14.5k33278
add a comment |Â
add a comment |Â
Student is a new contributor. Be nice, and check out our Code of Conduct.
Student is a new contributor. Be nice, and check out our Code of Conduct.
Student is a new contributor. Be nice, and check out our Code of Conduct.
Student is a new contributor. Be nice, and check out our Code of Conduct.
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Could you summarize the "realistic box problem" that you link to? That link requires Java in the browser, which many people (including myself) disable for security reasons.
– Rory Daulton
6 hours ago
@RoryDaulton: Here is that problem: "A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. If six congruent cuts (denoted in black) are to be made into the cardboard and five folds (denoted by dotted lines) made to adjoin the cuts so that the resulting piece of cardboard is to be folded to form a closeable rectangular box (see figures below). How should this be done to get a box of largest possible volume?"
– Joseph O'Rourke
6 hours ago
@JosephO'Rourke: Thanks! Is there any way to show the figures with the cuts and folds and final box?
– Rory Daulton
5 hours ago
1 covers a lot of ground: number of shale wells in a section, number of salespeople, hours to keep a store open, etc.
– guest
3 hours ago