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To find the thickness of the tape, we can use the given information about the total length of the tape, the outer diameter, and the inner diameter when the tape is rolled up.

First, we can calculate the cross-sectional area of the tape when it's rolled up by considering it as a cylindrical shell. The formula for the area of a cylindrical shell is given by the difference of the areas of two circles (the outer circle and the inner circle).

1. Convert diameters to radii in meters:

   [ \text{Outer radius (R)} = \frac{10 , \text{cm}}{2} = 5 , \text{cm} = 0.05 , \text{m} ]

   [ \text{Inner radius (r)} = \frac{5 , \text{cm}}{2} = 2.5 , \text{cm} = 0.025 , \text{m} ]

2. Calculate the area of the cross section:

   [ \text{Area of the cylindrical shell (A)} = \pi (R^2 - r^2) = \pi (0.05^2 - 0.025^2) ]

   [ A = \pi (0.0025 - 0.000625) = \pi \times 0.001875 \approx 0.00589 , \text{square meters} ]

3. Determine the thickness of the tape (t):

   Since the tape is 100 meters long and the cross-sectional area is (0.00589 , \text{m}^2), the volume (V) of the tape can be calculated using:

   [ V = \text{length} \times \text{thickness} = A \times t ]

   Solving for (t):

   [ t = \frac{V}{A} = \frac{\text{length}}{A} = \frac{100 , \text{m}}{0.00589 , \text{m}^2} ]

   [ t \approx 0.0016964 , \text{meters} ]

   Converting this thickness into millimeters:

   [ t \approx 0.0016964 , \text{m} \times 1000 = 1.6964 , \text{mm} ]

   Therefore, the thickness of the tape is approximately 1.70 mm.