Final answer:
Shannon's countertop has a length of 9 ft and an area of 27 square feet. The area of Marta's larger square is 4 times the area of the smaller one. For uncertainty in area measurements, you multiply the upper and lower bounds of length and width and then subtract the smallest possible area from the largest to find the uncertainty range.
Step-by-step explanation:
To find the length of Shannon's rectangular countertop, we multiply the width, which is 3 ft, by 3. So, the length is 3 ft × 3 = 9 ft. To calculate the area, we use the formula for the area of a rectangle, which is length × width. Thus, the area is 9 ft × 3 ft = 27 square feet.
In the example of Marta's squares, the side length of the larger square is 4 inches × 2 = 8 inches. Since the area of a square is side length squared, the area of the smaller square is 4 inches × 4 inches = 16 square inches, and the area of the larger square is 8 inches × 8 inches = 64 square inches. This shows that the area of the larger square is 4 times the area of the smaller square, since (2×4 inches)² = 4 × (4 inches)².
For the question on the uncertainty in measurements, the area of the rectangular room can be calculated by multiplying the measured length and width. If we have a length of 3.955 ± 0.005 m and a width of 3.050 ± 0.005 m, the central area value would be 3.955 m × 3.050 m = 12.07275 m². To calculate the uncertainty, we use the maximum possible values subtracted by the minimum possible values, which result from using the length and width at their respective upper and lower bounds.