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Biggest Box
How to Make a Box
Given a sheet of cardboard that measures 3ft x 4 ft, let’s create a box that yields the greatest volume. In order to build a box out of a rectangular sheet, we need to cut out congruent squares from each of the corners so that the four sides can fold up to generate a topless 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?
______.
- What is the width of the sheet after cutting out the squares?
______.
- The dimensions of the box is
______.
Investigating Box Data
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?
Height (h)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
Discuss any problems that you encountered.
______
______
______.
From the table, we see that we have limits on how big of a square can be cut out.
What is the interval?
______.
Investigating Maximization Using Algebra
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 for this graph? (Consider our interval).
______.
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? ______
______.
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 and 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 = ______and y = ______
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
______.
Investigating Extrema Using Calculus
Find the derivative of the following functions:
1. f(x) = 4x2 – 7x + 20
______.
2. y = (5x + 3)(2x – 1)
(You can do this 2 ways: FOIL first then find the derivative or use the Product Rule)
______.
3. f(z) = z(z -1)(z + 1)
______.
Now find the derivative of the function of our box.
4. f(x) = x(4 - 2x)(3 - 2x)
______.
[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. ______.
2. ______.
3. ______.
Now find the critical numbers for our box.
4. ______.
Can we eliminate any numbers? Think about our interval.
______.
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: ______.
│______│______│______│
2. critical number: ______.
│______│______│
3. critical numbers: ______.
│______│______│______│
Now let’s do it for the critical numbers of our box.
4. critical number: ______.
│______│______│
Conclusion
So is our critical point a local maximum or minimum? ______.
Then the dimensions of our box that will yield the greatest volume is:
______.
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. This is called defining. 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 à f1(x) (Fig. 10).
Fig.10
Now that we have our derivative function, how do we find the critical numbers?
______.
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 comma x. 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?
______.
What is our original interval?
______.
Fig.13
So the volume of our box with the optimal values for x is?
______.
Investigating the data by using the dynamic geometry software in the TI-Nspire
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 the graphing tool.
This is the first screen you will see after opening up the 2: 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?
Why is that?
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?
What is the minimum length we could cut from the corners of the box?
Why is that?
Then what is the minimum x could be in this case?
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:
To place a point on the line previously constructed, press menu, 6: Points & Lines, and then 2: Point On. Then simply select the point on the line where you would like your x to be (the point should be on the smaller line segment as well as the large one). You should see this:
After you draw that point hit esc, you should be able to drag that new point you just made along the line segment. IMPORTANT: You should NOT be able to drag this point any further than the endpoints of the small line segment, if you are able to do this then you did not place it on the correct line and need to do it again (press ctrl + Z to undo).