Final answer:
The longest diagonal in a box with dimensions 12 inches by 12 inches by 18 inches is calculated using the 3-dimensional Pythagorean theorem and approximates to 24.7 inches.
Step-by-step explanation:
To find the length of the longest diagonal in a rectangular box, we use the 3-dimensional version of the Pythagorean theorem. The box dimensions are 12 inches by 12 inches by 18 inches, often referred to as length, width, and height, respectively.
First, calculate the diagonal of the base (which is a square in this case). This is the diagonal across the square's face with sides of 12 inches:
base diagonal = √(12^2 + 12^2) = √(144 + 144) = √288 = 12√2 inches
Now, apply the Pythagorean theorem to find the longest (body) diagonal using the base diagonal and the height of the box (18 inches):
longest diagonal = √((base diagonal)^2 + height^2) = √((12√2)^2 + 18^2) = √(288 + 324) = √612
Finally, evaluate the square root:
√612 ≈ 24.7 inches
Thus, the length of the longest diagonal in the box is approximately 24.7 inches, which corresponds to option B.