(a) To find the power of the car's engine at point A, we first calculate the driving force using Newton's second law:

\(D - 500 = 1200 \times 0.8\)

\(D = 1460 \text{ N}\)

The power \(P\) is given by:

\(P = D \times v = 1460 \times 15 = 21900 \text{ W}\)

(b) To show that the distance AB is 1362.6 m, we first calculate the change in kinetic energy:

\(\text{Change in KE} = \frac{1}{2} \times 1200 \times 32^2 - \frac{1}{2} \times 1200 \times 15^2\)

\(= 614400 - 135000 = 479400 \text{ J}\)

The work done by the engine is:

\(\text{Work done} = 21900 \times 53 = 1160700 \text{ J}\)

Using the work-energy principle:

\(1160700 - 500 \times d = 479400\)

\(500d = 1160700 - 479400\)

\(500d = 681300\)

\(d = \frac{681300}{500} = 1362.6 \text{ m}\)