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Imagine constructing an open-topped water tank from a square metal sheet (2 m by 2 m). You would cut squares from the four corners of the sheet and bend up the four remaining rectangular pieces to form the sides of the tank. Then you would weld the edges together to make them watertight. Your goal is to construct a tank with the greatest possible volume.
What size squares do you conjecture would result in the water tank with the greatest volume?
To investigate the relationship between the maximum volume of the tank and the size of the squares cut from the corners, build models and collect data. Using a 1:10 scale, start with a model — 20-by-20-cm square sheets (PDF – be sure to print this document full scale) of paper. Take one sheet of paper and cut a 1-by-1-cm square from each corner. Fold the net into an open box and tape it. You have just constructed a scale model of the water tank. Repeat the process to construct different models of the water tank.
Problem C1
Collect data on the different-sized water tanks you can make. The side lengths of the cutout squares in centimeters must be whole-number values. Record your data by filling in the table below:
Size of the Cutout Square (cm) |
Dimensions of the Box (cm) |
Volume of the Box (cm^{3}) |
1 by 1 | 1 by 18 by 18 | 324 |
Problem C2
Using whole-number side lengths, which size of a cutout square results in the largest volume for the box? What is the size of the cutout square and the resulting volume?
Problem C3
These models were at a scale of 1:10. If the largest box you made represented the water tank, what would its dimensions be?
Problem C4
One way to get a closer estimate of the dimensions of the box with the greatest volume is to make a graph. On the x-axis, plot the length of the sides of the cutout square (in centimeters); on the y-axis, plot the corresponding volume of the box (in cubic centimeters). Use a graphing calculator, sketch your graph on grid paper (PDF file), or enter the data into the graphing program on your computer.
What would happen if you could remove squares from the corners that used decimals, such as side of square = 3.5 cm, or side = 3.75 cm? Approximate the size of the squares that should be cut to maximize the resulting volume. Note 3
Note 3
Graphing the data may help you realize that the side length of the square that produces the maximum volume is not a whole-number value. Also, while we can approximate the maximum volume of the water tank, the actual maximum is not easily determined by this method. To find the maximum volume more precisely, we would need to use calculus.
Problem C1
Size of the Cutout Square (cm) |
Dimensions of the Box (cm) |
Volume of the Box (cm^{3}) |
1 by 1 | 1 by 18 by 18 | 324 |
3 by 3 | 3 by 14 by 14 | 588 |
4 by 4 | 4 by 12 by 12 | 576 |
5 by 5 | 5 by 10 by 10 | 500 |
6 by 6 | 6 by 8 by 8 | 384 |
7 by 7 | 7 by 6 by 6 | 252 |
8 by 8 | 8 by 4 by 4 | 128 |
9 by 9 | 9 by 2 by 2 | 36 |
Problem C2
The largest volume seems to result from a 3-by-3 cutout square (588 cm^{3}). The 4-by-4 square gave nearly as high a volume.
Problem C3
You found that the largest tank would result if you removed 3-by-3 cm squares. The dimensions of the model would be 17 by 17 by 3 cm. Increasing back to the original scale, the dimensions of the tank would be 170 by 170 by 30 cm.
Problem C4
From observing the graph, it becomes evident that the largest value for volume will be between values 3 and 4 on the x-axis.
Problem C5
Using 3.5 as a square’s side would give us the volume of 591.5 cm^{3}. Using 3.4 as a square’s side, we’d get the volume of 592.4 cm^{3}. The largest volume is achieved when the square is cut with side length 3 1/3 (or 3.333…) cm, leaving 13 1/3 (or 13.333…) cm in the center. The volume is (3 1/3) • (13 1/3) • (13 1/3) = (10/3) • (40/3) • (40/3) = 16,000/27 cm^{3}, or about 592.59 cm^{3}.