1. **State the problem:** We have air initially at atmospheric pressure $P_1 = 100000$ Pa. It is pumped into a tyre where the pressure is $P_2 = 550000$ Pa and volume is $V_2 = 3000$ cm³. We want to find the volume $V_1$ of the air at atmospheric pressure. Then, the pressure of the air inside the tyre is increased to $P_3 = 600000$ Pa, and we want to find the new volume $V_3$ of air in the tyre. 2. **Use Boyle's Law:** Boyle's law states that for a fixed amount of gas at constant temperature: $$P_1 V_1 = P_2 V_2$$ From this, we can find $V_1$: $$V_1 = \frac{P_2 V_2}{P_1}$$ 3. **Calculate $V_1$: ** $$V_1 = \frac{550000 \times 3000}{100000} = 16500 \text{ cm}^3$$ So the volume of air at atmospheric pressure is $16500$ cm³. 4. **Find $V_3$ when the pressure increases to $P_3 = 600000$ Pa:** Here, the amount of air corresponds to the same volume at atmospheric pressure $V_1 = 16500$ cm³, so using Boyle's law again: $$P_1 V_1 = P_3 V_3$$ Solve for $V_3$: $$V_3 = \frac{P_1 V_1}{P_3} = \frac{100000 \times 16500}{600000} = 2750 \text{ cm}^3$$ 5. **Final answers:** - Volume at atmospheric pressure $V_1 = 16500$ cm³ - Volume at $600000$ Pa $V_3 = 2750$ cm³