Question

your solution will satisfy the following equations:$square \times square \times square = 30$$square^3 = 30$this work is licensed under a creative commons attribution - noncommercial-grade 8 - unit 5georgias k-12 mathematics standardsname: date:shipping dicedice (six-sided number cubes) come in many different sizes. three sizes are listed below by volume.size #1: $125 \text{mm}^3$ (extremely small)size #2: $512 \text{mm}^3$ (super small)size #3: $1,728 \text{mm}^3$ (small)1. design a box to hold and ship 1,000 extremely small dice with no space left within the box. what are the dimensions of your box? (length, width, and height)2. design a box to hold and ship 1,000 super small with no space left within the box. what are the dimensions of your box? (length, width, and height)3. design a box to hold and ship 1,000 small dice with no space left within the box. what are the dimensions of your box? (length, width, and height)

Explanation:

Step1: Find edge length of die #1

The volume of a cube is $V = s^3$, so solve for $s$:
$s_1 = \sqrt[3]{125} = 5$ mm

Step2: Calculate total volume for 1000 dice #1

Total volume $V_{total1} = 1000 \times 125 = 125000$ mm³

Step3: Find box dimensions for die #1

For a cube-shaped box (no leftover space), edge length $S_1 = \sqrt[3]{125000} = 50$ mm. This can also be calculated as $10 \times s_1 = 10 \times 5 = 50$ mm (arranging 10 dice along each axis).

Step4: Find edge length of die #2

$s_2 = \sqrt[3]{512} = 8$ mm

Step5: Calculate total volume for 1000 dice #2

$V_{total2} = 1000 \times 512 = 512000$ mm³

Step6: Find box dimensions for die #2

$S_2 = \sqrt[3]{512000} = 80$ mm, or $10 \times s_2 = 10 \times 8 = 80$ mm.

Step7: Find edge length of die #3

$s_3 = \sqrt[3]{1728} = 12$ mm

Step8: Calculate total volume for 1000 dice #3

$V_{total3} = 1000 \times 1728 = 1728000$ mm³

Step9: Find box dimensions for die #3

$S_3 = \sqrt[3]{1728000} = 120$ mm, or $10 \times s_3 = 10 \times 12 = 120$ mm.

Answer:

1. Length: 50 mm, Width: 50 mm, Height: 50 mm
2. Length: 80 mm, Width: 80 mm, Height: 80 mm
3. Length: 120 mm, Width: 120 mm, Height: 120 mm