To solve this problem, follow these steps:
1. Understand the relationship between the original dimensions and the new dimensions:
- The area of a rectangle is calculated as the product of its length and width.
- If both the length and the width of a rectangle are multiplied by a scale factor, the new area will be calculated based on the squared value of that scale factor.
2. Identify the given information:
- The initial area of the classroom is 420 square feet.
- The length and width of the classroom are both multiplied by 3.
3. Calculate the new area:
- When both dimensions (length and width) of a rectangle are scaled by the same factor, the area of the rectangle changes by the square of that factor.
- Here, the scale factor is 3. Therefore, the new area will be:
[tex]\[
\text{New Area} = \text{Original Area} \times (\text{Scale Factor})^2
\][/tex]
- Substitute the given values into the equation:
[tex]\[
\text{New Area} = 420 \, \text{square feet} \times (3)^2
\][/tex]
[tex]\[
\text{New Area} = 420 \, \text{square feet} \times 9
\][/tex]
- Perform the multiplication:
[tex]\[
\text{New Area} = 3780 \, \text{square feet}
\][/tex]
4. Compare the result with the possible answers:
- The calculated new area is 3780 square feet.
Therefore, the correct answer is:
D) [tex]$3780 \, \text{ft}^2$[/tex]
