To find the smallest diameter of the pipe that will fit the fiber optic line in the rectangular casing BCDE, we need to determine the distance across the rectangle.
The dimensions provided are:
- DE = 2.5 cm
- BE = 3 cm
Since it is a rectangle, we assume that the distances can represent the dimensions of BCDE where BC = DE and CD = BE. Therefore, the dimensions of rectangle BCDE are:
- Length (BC or DE) = 2.5 cm
- Width (BE or CD) = 3 cm
Next, we need to calculate the diagonal of the rectangle, which represents the longest distance across the rectangle. The diagonal \(d\) can be calculated using the Pythagorean theorem:
\[ d = \sqrt{(length)^2 + (width)^2} \]
Substituting in the dimensions:
\[ d = \sqrt{(2.5)^2 + (3)^2} \]
Calculating the squares:
\[ d = \sqrt{(6.25) + (9)} = \sqrt{15.25} \]
Now calculating the square root:
\[ d \approx 3.905 \]
To find the diameter of the pipe that can fit through this diagonal, we take the diagonal length as the diameter, as it is the maximum distance across.
Now, rounding to the nearest hundredth:
\[ d \approx 3.91 \text{ cm} \]
Thus, the smallest diameter of the pipe that will fit the fiber optic line is \( \boxed{3.91 \text{ cm}} \).
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