Solutions for Session 9, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5

Problem C1

Size of the
Cutout
Square (cm)

Dimensions
of the Box
(cm)

Volume
of the Box
(cm3)

 1 by 1 1 by 18 by 18 324 2 by 2 2 by 16 by 16 512 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 cm3). 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 cm3. Using 3.4 as a square's side, we'd get the volume of 592.4 cm3. 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 cm3, or about 592.59 cm3.

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