A test car is traveling along a level road at \(88 \mathrm{~km} / \mathrm{hr}\). In order to study the acceleration characteristics of a newly installed engine, the car accelerates at its maximum possible rate. The test crew records the following velocities at various locations along the level road:

A preliminary study shows that these data follow an equation of the form \(V=A\left(1+e^{B x}\right)\), where \(A\) and \(B\) are positive constants. Find \(A\) and \(B\) and a Lagrangian expression \(V=V\left(V_{0}, t\right)\), where \(V_{0}\) is the car velocity at time \(t=0\) when \(x=0\).

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x = 0 x = 0.1 km x = 0.2 km x = 0.3 km V = 88.5 km/hr V = 93.1 km/hr V = 98.3 km/hr x = 0.4 km V = 104.0 km/hr V = 110.3 km/hr x = 0.5 km x = 1.0 km V = 117.2 km/hr V = 164.5 km/hr.