Final answer:
The correct polynomial for the width of the river is 9x^2 - 16x, and when x = 3, the width is 33 feet. None of the options provided are correct.
Step-by-step explanation:
The polynomial that represents the width of the river between the two cabins is found by subtracting the distances from each cabin to the river from the total distance across the river. The total distance between the cabins is 12x^2 - 7x + 5 feet. Subtracting the distances of one cabin, which is 9x + 1 feet, and the other which is 3x^2 + 4 feet from the river, gives us:
Width of the river = (12x^2 - 7x + 5) - (9x + 1) - (3x^2 + 4)
Combine like terms:
Width of the river = 12x^2 - 7x + 5 - 9x - 1 - 3x^2 - 4
Width of the river = 9x^2 - 16x + 0
When x = 3, we calculate the width as follows:
Width of the river = (9(3)^2) - (16(3)) + 0
Width of the river = 81 - 48
Width of the river = 33 feet. None of the provided options (a, b, c, d) are correct.