A block of wood is a cube whose side is x in. long. You cut off a 1-inch thick piece from the entire right side. Then you cut off a 3-inch piece from the entire top of the remaining shape. The volume of the remaining block is 2,002 in^3. What are the dimensions of the original block of wood?
After cutting off a 1-inch thick piece from the right side, the remaining block will have a length of x - 1 inches.
Similarly, after cutting off a 3-inch piece from the top, the remaining block will have a width and height of x - 3 inches each.
The volume of the remaining block is given by (x - 1)(x - 3)(x - 3) = 2,002 in^3.
Expanding the equation, we have (x^2 - 4x + 3)(x - 3) = 2,002.
Multiplying (x - 3) to both terms in the parentheses gives us x^3 - 7x^2 + 15x - 6x^2 + 21x - 45 = 2,002.
Combining like terms, this simplifies to x^3 - 13x^2 + 36x - 45 = 2,002.
Next, move the constant term to the right side: x^3 - 13x^2 + 36x - 45 - 2,002 = 0.
This equation can be simplified to x^3 - 13x^2 + 36x - 2,047 = 0.
Notice that 23 is a factor of 2,047. Perform the synthetic division using 23 as the divisor to find the other factors: + 23 | 1 - 13 + 36 - 2,047
23 - 230 + 1,260
- 207 + 17 - 787
The result is x^2 + 23x - 207. Factoring this polynomial, we have (x - 9)(x + 23).
Therefore, the possible values for x are 9 and -23.
Since the dimensions cannot be negative, the original block of wood must have dimensions 9 inches by 9 inches by 9 inches. Answer: \boxed{9}.