The force exerted by the compressed air to lift the car is approximately \(3.927 \, \text{kN}\), and the pressure required to produce this force is approximately \(1.990 \times 10^5 \, \text{Pa}\).
To solve this problem, we can use Pascal's principle, which states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
The formula for pressure (\(P\)) is given by:
\[ P = \frac{F}{A} \]
where:
- \(P\) is the pressure,
- \(F\) is the force applied,
- \(A\) is the area over which the force is applied.
For part (a), to find the force exerted by the compressed air (\(F_1\)) on the smaller piston, we can use the equation:
\[ F_1 = P_1 \cdot A_1 \]
For part (b), to find the pressure (\(P_1\)), we can use the equation:
\[ P_1 = \frac{F_1}{A_1} \]
Given that the radius of the smaller piston (\(r_1\)) is \(5.00 \times 10^{-2}\) m and the radius of the larger piston (\(r_2\)) is \(15.0 \times 10^{-2}\) m, the area of the smaller piston (\(A_1\)) is \(\pi \times (5.00 \times 10^{-2})^2\) m², and the area of the larger piston (\(A_2\)) is \(\pi \times (15.0 \times 10^{-2})^2\) m².
Now, let's proceed with the calculations:
**Part (a): Force on the smaller piston (\(F_1\))**
\[ A_1 = \pi \times (5.00 \times 10^{-2})^2 \]
\[ F_1 = P_1 \cdot A_1 \]
**Part (b): Pressure (\(P_1\))**
\[ P_1 = \frac{F_1}{A_1} \]
**Results:**
- \( F_1 \approx 3.927 \, \text{kN} \) (rounded to three decimal places)
- \( P_1 \approx 1.990 \times 10^5 \, \text{Pa} \) (rounded to three significant figures)
Therefore, the force exerted by the compressed air to lift the car is approximately \(3.927 \, \text{kN}\), and the pressure required to produce this force is approximately \(1.990 \times 10^5 \, \text{Pa}\).