To solve this problem, let's go through it step-by-step.

Step 1: Determine the Area of the Floor

The floor is square with dimensions $7 \, \text{m} \times 7 \, \text{m}$.

$\text{Area of the floor} = 7 \times 7 = 49 \, \text{m}^2$

Step 2: Calculate the Area Covered by the Carpet

The carpet covers 90% of the floor area. So, the area covered by the carpet is:

$\text{Area of the carpet} = 0.9 \times 49 = 44.1 \, \text{m}^2$

Step 3: Find the Side Length of the Carpet

Since the carpet is square, let $s$ represent the side length of the carpet. The area of the carpet can then be written as:

$s^2 = 44.1$

Solving for $s$:

$s = \sqrt{44.1} \approx 6.64 \, \text{m}$

Thus, the dimensions of the carpet are approximately:

$6.64 \, \text{m} \times 6.64 \, \text{m}$

Step 4: Calculate the Area of the Floor Not Covered by the Carpet

The area of the floor not covered by the carpet is the difference between the total area of the floor and the area of the carpet:

$\text{Uncovered area} = 49 - 44.1 = 4.9 \, \text{m}^2$

Final Answers

1. Dimensions of the carpet: $6.64 \, \text{m} \times 6.64 \, \text{m}$
2. Area of the floor not covered by the carpet: $4.9 \, \text{m}^2$

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Would you like further clarification on any step?

Additional Questions to Consider:

1. What would the dimensions of the carpet be if it covered 80% of the floor?
2. How much area would remain uncovered if the carpet covered only 70% of the floor?
3. If the floor were rectangular (7 m by 5 m), how would that change the carpet's dimensions?
4. How much area would the carpet cover if it had dimensions of 6.64 m by 6.64 m on a larger floor?
5. How do we adjust the carpet size if we want only 5% of the floor uncovered?

Tip:

To quickly determine side lengths of squares when area percentages are involved, remember to take the square root after calculating the desired area.