The volume of the prism to the right is 315 cubic inches. Find the value of \( x \) and the dimensions of the prism.

We’re given that the volume of the prism is 315 cubic inches and that its dimensions are expressed in terms of x. A common way this type of problem is set up is to let the three dimensions be   x,   x + 2,   x – 2. Then the volume is given by   Volume = x · (x + 2) · (x – 2). Since the volume is 315 cubic inches, we have the equation   x · (x + 2) · (x – 2) = 315. Step 1. Simplify the Expression Notice that (x + 2)(x – 2) is a difference of squares, so   (x + 2)(x – 2) = x² – 4. This makes the volume equation become   x(x² – 4) = 315   x³ – 4x = 315. Step 2. Solve for x We need to solve   x³ – 4x – 315 = 0. One way to find a solution is by testing reasonable positive integer values (since dimensions must be positive). Let’s try x = 7:   7³ – 4·7 – 315 = 343 – 28 – 315 = 0. Since the equation equals 0 when x = 7, we have found that x = 7. Step 3. Find the Dimensions Now substitute x = 7 into the expressions for each dimension:   Dimension 1: x = 7 inches.   Dimension 2: x + 2 = 7 + 2 = 9 inches.   Dimension 3: x – 2 = 7 – 2 = 5 inches. Step 4. Verify the Volume Multiply the three dimensions to confirm the volume:   Volume = 7 × 9 × 5 = 315 cubic inches. This confirms our solution. Answer:   x = 7,   and the dimensions of the prism are 7 inches, 9 inches, and 5 inches.