Answer: To find x, we can use the formula for the volume of a rectangular prism: V = lwh, where l is the length, w is the width, and h is the height.
Substituting the given dimensions, we have:
V = (x + 6)(x - 1)(x - 2) = 60
Expanding the product on the left side, we get:
V = (x^2 + 4x - 12)(x - 2) = 60
Multiplying out the right side, we get:
V = x^3 - 2x^2 + 4x^2 - 8x - 60 = x^3 + 2x^2 - 8x - 60
Combining like terms, we have:
x^3 + 2x^2 - 8x - 60 - 60 = 0
Simplifying, we get:
x^3 + 2x^2 - 8x - 120 = 0
Now we can use various methods to solve for x. One possible method is to use synthetic division with the possible rational roots of ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±20, ±30, ±40, and ±60 (which are factors of the constant term 120) to narrow down the possibilities. Trying x = 2 as a root (since it is a factor of the constant term and also one of the dimensions of the prism), we get:
2 | 1 2 -8 -120
| 2 8 20
|--------------
| 1 4 0 -100
This means that x^3 + 2x^2 - 8x - 120 = (x - 2)(x^2 + 4x - 100), so the remaining quadratic factor can be factored as:
x^2 + 4x - 100 = 0
Using the quadratic formula, we get:
x = (-4 ± sqrt(4^2 - 4(1)(-100))) / (2(1))
x = (-4 ± sqrt(416)) / 2
x = (-4 ± 8sqrt(13)) / 2
x = -2 ± 4sqrt(13)
Since x represents a length, we discard the negative solution and conclude that:
x = -2 + 4sqrt(13)
Explanation: