The polynomial function for the volume of packets, given the height x, and the length being 15cm longer than the height, is V(x, w) = x × (x + 15) × w. Width w must satisfy the condition w ≤ 90 - 2x - 15.
To write a polynomial function representing the possible volumes of packets with given dimensions in terms of its height x, we first establish the relationship between the dimensions. According to the problem, the length of a packet is 15cm longer than its height, so if the height is x, the length is x + 15cm. Since the sum of length, width, and height must be less than or equal to 90cm, we have x + (x + 15cm) + width ≤ 90cm, simplifying to width ≤ 90cm - 2x - 15cm.
The volume V of a rectangular box is the product of its length, width, and height, so the volume function in terms of height x will be V(x) = x × (x + 15cm) × width. As we cannot determine the specific width from the given information, the function in terms of x and width w is V(x, w) = x × (x + 15) × w, where w must satisfy the condition w ≤ 90 - 2x - 15.