A piece of cardboard measures 12 inches by 10 inches is formed into an open top box by
cutting squares with side length x from each corner and folding up the sides.
Determine the volume function, the dimensions of the square cut from each corner to maximize
the volume, and the maximum volume.
We will be solving this problem graphically using a graphing calculator.
We begin with this 12 inch by 10 inch piece of cardboard.
Then we cut out these red squares from each corner, fold up the sides, and form this open
top box.
Notice when we fold up these edges, the height of the box is x inches.
this length here would be the length of this edge.
Notice how this would be a total of 10 inches minus x inches minus x inches or 10 minus
2x inches.
Let's go ahead and label this on the box, which would be this length here.
And then this length would be the length of this edge, which is 12 inches minus x inches
minus x inches, which gives us the length of 12 minus 2x inches . Let's label this edge
as well.
Because the volume is equal to length times width times height, this is the information
we need in order to write the volume function.
The volume function will be V of x because the input variable is x.
Let's change the order of this multiplication.
Let's write the volume as x times the quantity 10 minus 2x times the quantity 12 minus 2x.
Before we go to the next slide though, let's also talk about the domain of this volume
function.
We know we have to cut a square from each corner or the original 12 inch by 10 inch
piece of cardboard, and therefore x has to be greater than 0, but also notice how if
x is 5, this length would be 0 inches, which would not give us a open top box, which means
x must be greater than 0 and less than 5.
Using interval notation, we have an open parentheses 0 comma 5 closed parentheses.
This is important when we take a look at the graph of the volume function.
So this is the volume function, but let's also multiply this out to show it in expanded
form.
Let's first multiply the two binomials, which will give us 4 products, 1, 2, 3, and 4.
We have V of x equals x times the quantity, 10 times 12 equals 120, then we have 10 times
negative 2x, which is negative 20 x or minus 20 x.
And then we have negative 2 x times 12, which is negative 24 x or minus 24 x and then negative
2x times negative 2 x is positive 4 x squared.
We have plus 4 x squared.
Now we will combine like terms here and here.
We have V of x equals x times, let's also write the terms in descending order, let's
write the 4 x squared first and then we have negative 20 x minus 24 x, that's minus 44
x plus 120.
And now we will distribute the x.
We have V of x equals 4 x cubed minus 44 x squared plus 120 x.
So when the question asks for the volume function, we can give it in this form or expanded form.
And now we will graph the volume function to determine the size square that will give
us the maximum volume and determine the maximum volume.
With the graphing calculator on, we press y equals.
Let's enter the volume function this first form here.
We enter x times the quantity 10 minus 2 x times the quantity 12 minus 2 x.
Let's begin with the standard window by pressing Zoom 6 for Zstandard.
We will probably have to adjust the window but this is a nice place to start.
Remember the domain is the open interval from 0 to 5, so we are really only concerned about
the graph over this interval here.
Looking at the graph, we need to adjust the horizontal axis as well as the vertical axis.
We need to see higher on the graph, which means we need to increase the y maximum.
Let's press the window key.
Let's change the x minimum to negative 2 enter.
Let's leave the x max at 10 and the x scale at 1.
But let's go ahead and change the y min to -20 and the y max to 120.
And let's also change the y scale to 10.
And now let's press graph and see what we have.
We may have to come back and adjust the window more.
This window looks much better.
We are looking for this high point here, which will give us the dimensions of the square
as well as the maximum volume.
Because this is a high point, we will use the maximum feature of the calculator to determine
the coordinates of this point.
We press 2nd Trace for the calculation menu.
And then number 4 for maximum.
Notice how it is asking for the left bound.
We need to move the cursor to the left of the high point we are looking for.
So we press the left arrow until we are on the other side of the hill or to the left
of the high point.
Let's say here, enter.
And now it's asking for the right bound.
We now move the cursor to the right side of the high point.
Let's say approximately here, press enter.
It say guess, but can just press enter again.
And notice how the ordered pair of this point show below.
The x coordinate is approximately 1.8107 and the y coordinate or the volume function value
is approximately 96.7706.
Let's record this ordered pair.
The first value in the ordered pair is x, which gives us the length of the side of the
square cut from each corner that will maximize the volume and the second value is volume
or in this case the maximum volume.
Let's write these results as a complete sentence.
We will say squares cut from each corner with a side length of 1.8107 inches will give a
maximum volume of 96.7706 cubic inches.
Of course it should be noted these values have been rounded to 4 decimal places.
I hope you found this helpful.