A hand pump is being used to inflate a bicycle tire that has a gauge pressure of 40.0 psi. If the pump is a cylinder of length 18.0 in. with a cross- sectional area of 3.00 in. \(^{2},\) how far down must the piston be pushed before air will flow into the tire?

The tire has a gauge pressure of 40 psi. This means that we need at least a pressure of 40 psi inside the pump for the air to flow into the tire.

In order to use the SI unit system, we should convert the pressure from psi to pascals (Pa). We know that 1 psi = 6894.76 Pa. Therefore, the required pressure in the pump is (40 psi) * (6894.76 Pa/psi) = 275790.4 Pa.

Now that we have the pressure in pascals, we can calculate the required force applied on the piston to achieve this pressure using the formula P = F/A. Rearranging the formula: F = P * A. The cross-sectional area of the pump is given in square inches, so we should first convert it to square meters: A = 3.00 square inches to square meters = 3.00 * (0.0254)^2 = 0.0019372 m^2. Now we can calculate the required force: F = P * A = 275790.4 Pa * 0.0019372 m^2 = 534.089 N.

To find out how far the piston must be pushed, we can use the concept of work. The work equation is W = F * d, where W is the work done, F is the force, and d is the distance. By rearranging the equation, we can get d = W / F. Since we want to find out how far the piston must be pushed, we need to find the value of W. The work required to apply the force is equal to the increase in potential energy of the air inside the pump, which is given by W = P * V, where P is the pressure and V is the volume of the pump. We know the required pressure inside the pump (P = 275790.4 Pa) and the length and cross-sectional area of the cylinder (dimensions of the pump). The volume of the cylinder can be calculated as V = A * L, where L is the length of the cylinder. To use SI units, convert the length of the cylinder to meters: 18 inches in meters = 18 * 0.0254 = 0.4572 m. Now we can find the volume V = A * L = 0.0019372 m^2 * 0.4572 m = 0.000885 m^3.

Now that we have the volume, we can find the work required to apply the force: W = P * V = 275790.4 Pa * 0.000885 m^3 = 244.124 J. Finally, we can calculate the distance the piston must be pushed: d = W / F = 244.124 J / 534.089 N = 0.4572 m. Now convert the distance from meters to inches: d = 0.4572 m in inches = 0.4572 * (1/0.0254) = 18 inches. So, the piston must be pushed down 18 inches before air will flow into the tire.