$$2C_4H_{10} + 13O_2 \rightarrow 8CO_2 + 10H_2O$$

From the equation, we can see that 2 moles of butane produce 8 moles of carbon dioxide. The molar mass of butane is 58.12 g/mol, and the molar mass of carbon dioxide is 44.01 g/mol. So, the mass of butane needed to produce 96.3 g of carbon dioxide is:

$$96.3 g CO_2 \times \frac{2 mol C_4H_{10}}{8 mol CO_2} \times \frac{58.12 g C_4H_{10}}{1 mol C_4H_{10}} = 147.3 g C_4H_{10}$$

Therefore, 147.3 g of butane is needed to produce 96.3 g of carbon dioxide.