1
Biggest Box
How to Make a Box
The students are presented with a problem that asks them to maximize the volume of a box constructed from a 3’x4’ sheet of cardboard. The only way to construct a box is to cut uniform squares from each of the four corners. Introduce to the students the concept of constructing a box.
Find the Dimensions
If we denote the cut out squares to be x ft by x ft:
- What is the length of the sheet after cutting out the squares?
______4-2x ft______.
- What is the width of the sheet after cutting out the squares?
______3-3x ft______.
- The dimensions of the box are
______(4-2x) by (3-2x) by (x) ft _____.
We allow the students to come up with a formula for the volume of a rectangular box.
Investigating Box Data
Teachers’s Note:
Goals of the activity: This method invites the student to really think about the importance of the data that we need to solve the problem. They are required to find the appropriate data that will aid them in understanding why the data is significant to our problem as well as finding the answer that we’re looking for. By the end of this lesson, the students will know how to:
- Communicate what they believe is the relevant data
- Use a table to see a pattern in our data
- Understand how the data will affect our problem solving
- Discard irrelevant data
Pre-Required Knowledge: The students will need to how to solve an algebraic equation. This will include knowing how to set up the equation as well.
Materials Needed: The only material required is a writing tool (pen or pencil).
We have a cardboard sheet with a length of 4 ft and width of 3 ft. In the table, choose the size of the cut out square. Then find the corresponding length, width, and volume. Try using decimal values for the height. Do you foresee any problems?
x / Length
(l) / Width
(w) / Volume
(h)(l)(w)
Example / .5 / 3 / 2 / 3
Box 1
Box 2
Box 3
Box 4
Box 5
Box 6
We let the students guess and check various values for the length of a square to be cut out so that students can get a feel for what we’re looking for. By letting them use the table, they begin to see that certain values cannot be used because the volume becomes a negative, which is impossible.
Discuss any problems that you encountered.
______We have to be careful when choosing values for x because we are
restricted to 1.5 as the maximum. The reason is that values greater than 1.5 will
an overlap when cutting. We can’t cut something out when it’s already been
cut.
From the table, we see that we have limits on how big of a square can be cut out.
What is the interval?
______0 < x < 1.5______.
Investigating Maximization Using Algebra
Teachers’s Note:
Goals of the activity: This investigation using algebra will allow the students to use the TI-Nspire calculators to manipulate our data to find the maximum or minimum. They will do this by looking at the graph of the equation and using the graphing calculator to aid them in their findings. By the end of this lesson, the students will know how to:
- Graph our problem equation using the TI-Nspire
- Interpret the graph so that they find the local extrema.
- Determine the appropriate interval for our particular problem.
Pre-Required Knowledge: The students will need to know how to follow directions for using the TI-Nspire. They will need to draw upon their prior knowledge of graphing to understand what the picture is telling them. They will also use any knowledge of other graphing calculators (not necessary).
Materials Needed: A TI-Nspire should be available to every student (or group of students) as well as a writing implement.
Using the TI-Nspire for this part of the investigation will be extremely beneficial. From the previous section, students understand how to find the volume. Now, using technology, we want the students to be able to analyze a visual representation without being hindered by computations.
We have the function for the dimensions of the box:
f(x) = x(4 – 2x)(3 – 2x)
Let’s use the TI-Nspire to see the local maximum and minimum of our function.
First turn on the Nspire and open a new document. Push the home button and choose option 2: Graphs & Geometry Application. (Fig.1)
Fig.1
Now let’s input our function f(x) = x times (4-2x) times (3-2x). Remember to put the multiplication symbol between x and the first set of parentheses and the second set. Then hit enter. (Fig.2)
Fig.2
What is the appropriate window size to obtain a complete graph? (Consider our interval).
______[0,5] , [-2,5]______.
For our equation, f(x) represents the volume of the box. According to the graph, we are only interested in the positive values of y. Why? __Negative y-values mean that the volume of the box will be negative which is not possible.
So let’s change the window size of our graph for a more appropriate setting. Push the menu button and choose option 4: Window and then choose option 3: Zoom – In (Fig.3).We could set a window size manually from option 1: Window Settings but we’ll use 3 for this instance. Push enter. Push enter again to zoom in even more. (Fig.4)
Fig. 3
Fig.4
According to the graph, where is the local maximum, approximately?
x = _____.6______and y = ______3______
Now we can use the Trace function of the calculator to see where the graph has a local maximum. Push the menu button and choose 5: Trace and then choose option 1: Graph Trace (Fig. 4a).
Fig.4a
Now using the arrow buttons, move the cursor towards the local maximum and stop when the screen shows an M with a box around it. This means that we have located a local maximum (Fig. 4b). Now push enter to place a point here and we can label the point. (Note: this must be done immediately after placing a point). Let’s call it A.
Fig.4b
Notice that the numbers are all mixed together with the function. Let’s clear up this mess. Hit the esc button to get out of our current function of Trace (the icon on the upper left of the screen should disappear). Move your cursor to the function statement until it turns into a hand (Fig. 4c).
Fig.4c
Now push and hold the middle button on your arrows wheel until hand grabs the label. Once this is done, move the label away from our area of interest (Fig.4d). Push the middle button again to release.
Fig.4d
Now we can see the coordinate of our local maximum. This is what we need to know to find our volume. So the dimensions of our box with the greatest volume is
______.57 by 2.86 by 1.86 ft.______.
Investigating Extrema Using Calculus
Teachers’s Note:
Goals of the activity: This method of using calculus will familiarize the students with using the TI-Nspire to solve an optimization problem. By the end of this lesson, the students will know how to:
- Store data into the TI-Nspire
- Find a derivative of an equation
- Find the critical numbers
- Use the derivative to find local extrema using the first derivative test.
Pre-Required Knowledge: The students will need to know the concept of the first derivative. A refresher will be given. They will need basic skills using the TI-Nspire but this will be provided as well. Computing critical numbers will be necessary as well.
Materials Needed: A TI-Nspire should be available to every student (or group of students). They will also need a pen or pencil.
Here, we want the students to rely on the education that they received in their pre-calculus/calculus courses to determine the extreme of our equation. We have them use the first derivative test with critical numbers. We could also have them extend on this to use the second derivative test but we felt that using the first derivative test was sufficient for what we were looking for.
Find the derivative of the following functions:
1. f(x) = 4x2 – 7x + 20
______(3x2 – 7)______.
2. y = (5x+ 3)(2x – 1)
(You can do this 2 ways: FOIL first then find the derivative or use the Product Rule)
______(20x + 1)______.
3. f(z) = z(z -1)(z + 1)
______(3z2 - 1)______.
Now find the derivative of the function of our box.
4. f(x) = x(4 - 2x)(3 - 2x)
______(12x2 – 28x + 12)______.
[1]Definition: A critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c) does not exist.
Now find the critical numbers of the derivative of the functions of ex. 1-3. Set the derivative that you found earlier equal to zero and solve for x.
1. ______[+/- (√21)/3]______.
2. ______(-1/20)______.
3. ______[+/- (√3)/3]______.
Now find the critical numbers for our box.
4. ______[(7(+/-)√13)/6]______.
Can we eliminate any numbers? Think about our interval.
____(eliminate (7+√13)/6)______.
Now that we have our critical number, we can find the local maximum. We don’t want a minimum because we are trying to maximize the volume of the box. Let’s use the first derivative test.
[2]The First Derivative Test: Suppose c is a critical number of a given function f.
a. If f’ goes from (+) to (-), then f has a local maximum at c.
b. If f’ goes from (-) to (+), then f has a local minimum at c.
c. If f’ has no sign change at c, then there is no local extrema.
Label the number line with the critical numbers and intervals from ex. 1-3. Then decide if there is a local maximum or minimum at each critical point.
1. critical numbers: ______[+/- (√21)/3]___
│______│______│______│
-∞-(√21)/3 (√21)/3 ∞
2. critical number: ______-1/20______.
│______│______│
-∞ -1/20 ∞
3. critical numbers: ______.
│______│______│______│
-∞-(√3)/3 (√3)/3 ∞
Now let’s do it for the critical numbers of our box.
4. critical number: ______7-√13)/6______.
│______│______│
-∞ 7-√13)/6 ∞
Conclusion
So is our critical point a local maximum or minimum? ______maximum ______.
Then the dimensions of our box that will yield the greatest volume is:
______(.57 x 2.86 x 1.86 estimation)______.
Now let’s use our Nspire calculator to do the same thing. Turn on the calculator and push the home button and choose option 1: Calculator (Fig.5).
Fig.5
We want to first store our function into the calculator memory. We do that by usingdefine. Push the menu button and choose option 1: Actions and then choose option 1: Define (Fig.6). Push enter.
Fig.6
Type in the function that we want to store/define including the f(x) and push enter (Fig.7). The Done statement means that we have officially stored the information into that calculator memory.
Fig.7
In order to find the local maximum, we need to work with the first derivative. So we need to define another function for the derivative, which we’ll call f1(x).
Push the menu button again and choose 1: Actions and then 1: Define. Push enter. Type our derivative function name f1(x) and push =. DO NOT PUSH ENTER YET.
Now we want the derivative of our original function. Push the menu button and choose 5: Calculus and then 1:Derivative. This will bring up a template that we can fill in (Fig. 8).
Fig.8
We want to enter x in the denominator and then tab over to the larger set of parentheses. Here is where we input the original function, just the name. So type in f(x). Now push enter. Once again, you should see Done, which confirms our input (Fig.9).
Fig.9
Now we need to find the derivative. Type in the name of our derivative function and push enter f1(x) (Fig. 10).
Fig.10
Now that we have our derivative function, how do we find the critical numbers?
______Set the derivative equation to zero______.
Now push the menu button and choose 4: Algebra and then 1: Solve (Fig.11).
Fig.11
Here, we want to find the critical numbers. So type the name of our derivative, set = 0, and then commax. This means that we’re solving the derivative function = 0 for the variable x (Fig.12). Push enter.
Fig.12
Once again, it gives us two critical numbers (Fig.13). However, we can eliminate one. Why?
The larger critical number will put us out of our interval.
What is our original interval?
______0 < x < 1.5 ft______.
Fig.13
So the volume of our box with the optimal values for x is?
______(.57 x 2.86 x 1.86 estimation)______.
Investigating the data by using the dynamic geometry software on the TI-Nspire
Teachers Note:
Goals of the activity: This inquiry-based activity is geared towards the students learning how to use the TI-Nspire to solve an optimization problem that they are faced with. By the end of this lesson, the students will know how to:
- Draw a picture using the dynamic geometry software on the Nspire
- Connect that picture to a formula on the graphing software on the Nspire
- Use that graph to collect data from the picture and transfer it to a spreadsheet to analyze the results
- Analyze the outcomes of the spreadsheet to determine the maximum amount of volume one can achieve by using a single sheet of cardboard
Pre-Required Knowledge: The students will need to know the basis concept of optimization, the formula for finding the volume of a box, and basic knowledge of how to use the TI-Nspire (although much of it is covered in this lesson).
Materials Needed: A TI-Nspire should be available to every student (or group of students) to use for this problem, and they each need a pencil.
Our problem is the following:A sheet of cardboard 3 feet by 4 feet will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with the largest volume?
We have already looked at other ways to solve this, but now we are going to use applications in the TI-Nspire that will help us solve this problem by drawing a picture of the situation, graphing the possible outcomes, converting the information to a table, and analyzing it to find our maximum. The following instructions will walk you through how to go about this process.
Turn on the Nspire and hit the home button. From there, choose 2: Graphs & Geometry to open up thegraphing tool. /
This is the first screen you will see after opening up the Graphs & Geometry application:
To hide the axis and entry line (because we will not be using them for now) press menu, then 2: View, and select 1: Hide Axes. Press menu again, then 2: View, and select 3: Hide Entry Line.
The goal here is to draw two perpendicular rectangles to represent the cardboard after the corners have been cut, so the final product should look something like this:
(You will need to refer back to this drawing so you may choose to tear out this sheet and keep it aside)
Which is the computerized representation for this:
Where x represents the amount cut off of each corner to fold up the box.
To begin drawing the layout of the cardboard on the Nspire press menu, then 6: Points & Lines, and finally 5: Segment to create the first line segment.
Place the cursor where you would like the line to begin (preferably in the middle of the screen and work to the right of this beginning line segment) and press the click button to place the beginning of the line segment there. Then drag the cursor across the screen until the segment is at the desired length (only go about halfway across the screen to make sure you have left room for the graphing portion).
What is the maximum length we could cut the corners of the cardboard box? 1.5 feet
Why is that?You can’t cut more than halfway to 3 feet (the smallest side) because then you will be overlapping what you are cutting and that is impossible to do.
In our construction, x is the amount we are cutting off of the corners of the rectangular sheet of cardboard; if this is the case then what is the maximum amount x could possibly be? x 1.5
What is the minimum length we could cut from the corners of the box?0 feet
Why is that?Because if you don’t cut anything off the corners of the box then you have no sides to fold up thus resulting in zero for the height and your volume to be zero as well.
Then what is the minimum x could be in this case? X > 0
Therefore, we need to set a boundary on x so it can be no greater than 1.5 but no less than 0. To do this we will construct a separate line segment that measures exactly 1.5. Construct the line segment just as before, selecting menu, 6: Points & Lines, then 5: Segment. Place your cursor underneath the line you just created and click to create another line.
To make sure that this line is no longer than 1.5 feet (or cm, because that is what our calculator measures in so we will just substitute these units in for feet for the time being) use the measurement application. Press menu, 7: Measurement, and 1: length. Click one endpoint of the small segment, then the other endpoint, position the text where you want it and click once more.
Unless you are a really great estimator, then you will probably notice that the measurement doesn’t read 1.5 cm like we want it to. To fix this, press esc to get out of the measurement application and then click one endpoint and drag it until it reads 1.5 cm (you may need to make it bigger or in some cases smaller).
This will be our base line, because we know that x cannot be smaller than 0 cm or greater than 1.5 cm. Now, to make sure that we don’t cut our corners any bigger than this line place your cursor over the line and click the click button and hold it until you see a hand grabbing it. Drag the small line segment up on top of the large line segment and align the two right endpoints and click to let go. Your line-on-line segment should look like this: