Let's solve this step by step.

The original rectangular carpet has dimensions of 6 feet by 8 feet.

The problem mentions a "larger rectangular piece of carpet" that has dimensions that are "times longer." Since the word "times" suggests multiplication, but no specific multiplier is given, I'll assume you're asking for a general solution based on some multiplier $x$, which we'll substitute later if needed.

Step 1: Area of the original carpet

The area $A$ of a rectangle is calculated by multiplying its length and width: $A = \text{length} \times \text{width} = 6 \times 8 = 48 \text{ square feet}.$

Step 2: Dimensions of the larger carpet

If the larger carpet is $x$ times longer, both dimensions (length and width) will be multiplied by $x$. So, the new dimensions will be: $\text{New length} = 6x, \quad \text{New width} = 8x.$

Step 3: Area of the larger carpet

The area of the larger carpet is: $A_{\text{larger}} = (\text{new length}) \times (\text{new width}) = (6x) \times (8x) = 48x^2 \text{ square feet}.$

Conclusion:

The area of the larger carpet is $48x^2$ square feet, where $x$ is the factor by which the dimensions are scaled.

If you provide the specific multiplier $x$, I can give you the exact area.

Would you like any additional details or clarifications?

Here are 5 related questions for further exploration:

1. If the larger carpet's dimensions are 3 times longer, what is the area?
2. What is the perimeter of the original carpet?
3. How much larger is the area of the bigger carpet compared to the original?
4. If the larger carpet costs $5 per square foot, what would be the total cost?
5. What would be the diagonal length of the original and larger carpets?

Tip: When scaling dimensions of a rectangle, the area increases by the square of the scaling factor.