To determine if Elena's table can fit the three cube-shaped cases, we first need to calculate the dimensions of each case and compare the area required to the table's surface area.

Step 1: Calculate the side length of each cube-shaped case

The volume $V$ of a cube is given by the formula:

$V = s^3$

where $s$ is the side length of the cube. Since the volume of each case is 1,000 cubic inches, we can solve for $s$:

$1000 = s^3$

Taking the cube root of both sides:

$s = \sqrt[3]{1000} = 10 \text{ inches}$

So, each cube-shaped case has side lengths of 10 inches.

Step 2: Find the area required for the three cases

Each case has a footprint (area of the base) equal to:

$\text{Area of one case} = s \times s = 10 \times 10 = 100 \text{ square inches}$

For three cases, the total area required would be:

$\text{Total area} = 3 \times 100 = 300 \text{ square inches}$

Step 3: Convert the table dimensions to inches

The table dimensions are given in feet, so we need to convert them to inches (since 1 foot = 12 inches):

$\text{Table length} = 212 \text{ feet} \times 12 = 2,544 \text{ inches}$ $\text{Table width} = 112 \text{ feet} \times 12 = 1,344 \text{ inches}$

The surface area of the table is then:

$\text{Table area} = 2,544 \times 1,344 = 3,417,216 \text{ square inches}$

Step 4: Compare the areas

The table's area (3,417,216 square inches) is much larger than the area required for the three cases (300 square inches).

Conclusion:

Yes, Elena can use this table, as it is large enough to fit the three cube-shaped cases easily. In fact, it provides much more space than is needed.

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Would you like further details on any part of this explanation? Here are five questions that expand on this topic:

1. How would the result change if the cases were rectangular instead of cubes?
2. What if the cases were stacked vertically—how much height would be required?
3. How would you calculate the area needed if there were 10 cases instead of 3?
4. How does the relationship between volume and surface area differ for different shapes?
5. How could you arrange the cases on the table to optimize space usage?

Tip: When working with cube-shaped objects, the cube root is a key tool to move between volume and side length!