Max Volume

Andy Norton

Department of Mathematics Education

University of Georgia

Maximizing Volume

Our assignment is to find the maximum volume of a box created by cuting squares from each corner of a 25x15 inch rectangle and folding up the sides.

First, we should find a formula for the volume of our box. We know that:

Volume = Height*Width*Length

So, if we cut X inches from our rectangle:

V = (15-2X)(25-2X)X.

Suppose, then, we want to know when the volume will be 400 cubic inches. We set V=400 and solve for X. We get an equation which can be set equal to zero. We can solve this equation by using algebra. However, it might be simpler to use GSP. You can click on the picture above and observe an animation.

Note: The sketch in the GSP sketch is scaled down by a factor of 5, the resulting volumes, then, will be reduced by a factor of 5 cubed, or 125.

Let's return to the problem of finding a maximum volume. We can use Calculus to find this value. The idea is to take the derivative of our volume expression with respect to X. This will relate the rate of change in volume for any X. When this rate is zero, we know that we have reached a relative maximum or minumum point.

We can go on to solve this equation using the quadratic formula. We should get X = 3.9 inches.

Again, though, there are other, perhaps more revealing if not more accurtate, ways to demonstrate this. We look at three programs:

Algebra Expresser

GSP

Excel

We have already seen a GSP sketch. By running the animation we can see where the volume is maximed. Now, let's turn to another graph:

The graph above was generated by Algebra Expresser from our equation for Volume. Volume is represented on the y-axis for varying x values. We can see where the volume reaches 400 and where the volume is maximized.

Here is an Excel spreadsheet demonstrating the same concept. We have the Volume and change in Volume (dV/dX) for each X value.

X	Volume	dV/dX
0	0	375
0.1	36.704	359.12
0.2	71.832	343.48
0.3	105.408	328.08
0.4	137.456	312.92
0.5	168	298
0.6	197.064	283.32
0.7	224.672	268.88
0.8	250.848	254.68
0.9	275.616	240.72
1	299	227
1.1	321.024	213.52
1.2	341.712	200.28
1.3	361.088	187.28
1.4	379.176	174.52
1.5	396	162
1.6	411.584	149.72
1.7	425.952	137.68
1.8	439.128	125.88
1.9	451.136	114.32
2	462	103
2.1	471.744	91.92
2.2	480.392	81.08
2.3	487.968	70.48
2.4	494.496	60.12
2.5	500	50
2.6	504.504	40.12
2.7	508.032	30.48
2.8	510.608	21.08
2.9	512.256	11.92
3	513	3
3.1	512.864	-5.68
3.2	511.872	-14.12
3.3	510.048	-22.32
3.4	507.416	-30.28
3.5	504	-38
3.6	499.824	-45.48
3.7	494.912	-52.72
3.8	489.288	-59.72
3.9	482.976	-66.48
4	476	-73
4.1	468.384	-79.28
4.2	460.152	-85.32
4.3	451.328	-91.12
4.4	441.936	-96.68
4.5	432	-102
4.6	421.544	-107.08
4.7	410.592	-111.92
4.8	399.168	-116.52
4.9	387.296	-120.88
5	375	-125
5.1	362.304	-128.88
5.2	349.232	-132.52
5.3	335.808	-135.92
5.4	322.056	-139.08
5.5	308	-142
5.6	293.664	-144.68
5.7	279.072	-147.12
5.8	26

We can see that the maximum volume is at X=3. Notice that we reach a maximum volume when dV/dX is zero.

Excel can also create a graph to demonstrate our change in volume:

This graph is similar to the one created by Algebra Expresser and models the same solutions. Such programs as GSP, Expresser and Excel can reinforce studies in Algrebra, Geometry and Calculus, as well as add meaning.