The volume of a rectangular prism is given by the product of its length, width, and height, which is V = l * w * h, where l is the length, w is the width, and h is the height.
We are given that the volume of the prism is 3w^3 + 14w^2 + 8w. We can use this information to find the height of the prism by dividing the volume by the product of the length and width.
We know that the length of the prism is 2 more than three times the width (l = 3w + 2). Therefore, the product of the length and width is l*w = (3w + 2) * w = 3w^2 + 2w
Now we can divide the volume by the product of the length and width to find the height of the prism:
V / (l * w) = (3w^3 + 14w^2 + 8w) / (3w^2 + 2w) = 3w + 14 + 8/3w^2 + 2w
So, the height of the prism is 3w + 14 + 8/3w^2 + 2w
Which is a simplified expression for the height of the prism, in terms of the width.
