The volume of a rectangular prism is 2x^3+5x^2+x-2. The height of the prism is 2x-1, and the length of the prism is x+2. Find the width of the prism.
w(2x-1)(x+2) = 2x^2 + 5x^2 + x - 2
w(2x-1)(x+2) = (2x-1)(x+2)(x+1)
w = x+1
To find the width of the rectangular prism, we need to divide the volume of the prism by the product of its length and height.
Given:
Volume = 2x^3 + 5x^2 + x - 2
Height = 2x - 1
Length = x + 2
The formula for the volume of a rectangular prism is:
Volume = Length * Width * Height
So we have the equation:
2x^3 + 5x^2 + x - 2 = (x + 2) * Width * (2x - 1)
To find the width, we can rearrange the equation and isolate it:
1. Distribute the factors on the right side:
2x^3 + 5x^2 + x - 2 = (2x - 1)(x + 2) * Width
2. Divide both sides by (2x - 1)(x + 2):
Width = (2x^3 + 5x^2 + x - 2) / ((2x - 1)(x + 2))
Now, let's simplify the expression:
Width = (2x^3 + 5x^2 + x - 2) / ((2x - 1)(x + 2))
To simplify further, we can use polynomial long division or synthetic division to divide (2x^3 + 5x^2 + x - 2) by ((2x - 1)(x + 2)). However, it seems difficult in this case to find an exact expression for the width without performing the division.
Therefore, the width of the rectangular prism is given by the expression:
Width = (2x^3 + 5x^2 + x - 2) / ((2x - 1)(x + 2))