Final answer:
The dimensions of the rectangle with an area of 247 sq km, where the length is 6 km longer than the width, are found by solving the quadratic equation w2 + 6w - 247 = 0, resulting in a width of 13 km and a length of 19 km. The area of a square doubled in size will have an area four times the original, as shown in Marta's square problem.
Step-by-step explanation:
To find the dimensions of the rectangle where its area is 247 sq km and the length is 6 km longer than the width, we can set up an equation with width represented as w and length as w + 6 km. The area of a rectangle is length × width, so (w + 6 km)×w = 247 sq km. Simplifying, we have:
w2 + 6w - 247 = 0.
Now we need to solve this quadratic equation. Factoring the quadratic, we find that (w + 19)(w - 13) = 0. That means w could be -19 or 13. Since width can't be negative, we conclude that the width is 13 km, making the length 19 km because it is 6 km longer than the width. Therefore, the correct answer is that the length of the rectangle is 19 km and the width is 13 km.
Now let's look into the Marta's squares question. If Marta has a square with side length of 4 inches, and the larger square has dimensions that are twice the first square, the side length of the larger square is 4 inches × 2 = 8 inches. Since the area of a square is calculated as side length2, the smaller square's area is 4 inches2 = 16 square inches. The area of the larger square is 8 inches2 = 64 square inches. Comparing the two areas, we see that the area of the larger square is 64 / 16 = 4 times the area of the smaller square.