Final answer:
To find the smallest diameter of a pipe to fit the fiber optic lines, calculate the diagonal of the rectangular casing using the Pythagorean theorem. The diagonal, which is about 4.61 cm, represents the required pipe diameter.
Step-by-step explanation:
To determine the smallest diameter of the pipe that will fit the fiber optic line housed in a rectangular casing BCDE with dimensions DE = 3 cm and BE = 3.5 cm, we must find the length of the diagonal of the rectangle, as this will be the required diameter to house the rectangle without any bending of the lines.
The rectangle BCDE has two sides DE and BE that we can consider as the lengths of a right-angled triangle, with the diagonal being the hypotenuse. The Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, can be used to find the hypotenuse.
Using the Pythagorean theorem:
Hypotenuse2 = DE2 + BE2
Hypotenuse2 = 32 + 3.52
Hypotenuse2 = 9 + 12.25
Hypotenuse2 = 21.25
Hypotenuse = √21.25
Upon calculating the square root of 21.25, we find the hypotenuse; our desired diameter to be approximately 4.61 cm. Therefore, the smallest diameter of the pipe that will fit the fiber optic line is 4.61 cm, which corresponds to option c).