To calculate the apparent cubic expansion of the mercury, we use the formula for apparent expansion:

\(\text{Apparent Expansion} = V_0 \cdot \left( \beta_m - \beta_g \right) \cdot \Delta T\)

Where:
- \( V_0 = 100 \, \text{cm}^3 \) (initial volume of mercury)
- \( \beta_m = 1.82 \times 10^{-4} \, \text{K}^{-1} \) (real cubic expansivity of mercury)
- \( \beta_g = 2.4 \times 10^{-5} \, \text{K}^{-1} \) (cubic expansivity of glass)
- \( \Delta T = 100^\circ C - 10^\circ C = 90 \, \text{K} \) (change in temperature)

First, we calculate \( \beta_m - \beta_g \):

\(\beta_m - \beta_g = 1.82 \times 10^{-4} - 2.4 \times 10^{-5} = 1.58 \times 10^{-4} \, \text{K}^{-1}\))

Now substitute the values into the apparent expansion formula:

\(\text{Apparent Expansion} = 100 \, \text{cm}^3 \cdot (1.58 \times 10^{-4}) \cdot 90\)

Calculating this gives:

\(= 100 \cdot 1.58 \times 10^{-4} \cdot 90\)

\(= 100 \cdot 1.422 \times 10^{-2}\)

\(= 1.422 \, \text{cm}^3\)

Thus, the apparent cubic expansion of the mercury is approximately:

\(\text{Answer: } 1.42 \, \text{cm}^3\)