An automated logistics drone starts an operation at Cycle k = 0 with an initial battery ch… — GMAT

An automated logistics drone starts an operation at Cycle $k = 0$ with an initial battery charge of $C_0 = 100\%$ and a payload mass of $P_0 = 40\text{ kg}$ . At the end of each subsequent cycle $k$ (where $k = 1, 2, 3, 4$ ), the payload mass decreases by $8\text{ kg}$ according to $P_k = P_{k-1} - 8$ , and the remaining battery charge updates according to the state transition rule $C_k = C_{k-1} - (5 + 0.2 P_{k-1})\%$ . Match each operational cycle on the left with its exact state $(C_k, P_k)$ at the end of that cycle on the right.

End of Cycle 1 $C_1 = 87.0\%, P_1 = 32\text{ kg}$
End of Cycle 2 $C_2 = 75.6\%, P_2 = 24\text{ kg}$
End of Cycle 3 $C_3 = 65.8\%, P_3 = 16\text{ kg}$
End of Cycle 4 $C_4 = 57.6\%, P_4 = 8\text{ kg}$

Cevap

End of Cycle 1 matches with C_1 = 87.0%, P_1 = 32 kg; End of Cycle 2 matches with C_2 = 75.6%, P_2 = 24 kg; End of Cycle 3 matches with C_3 = 65.8%, P_3 = 16 kg; End of Cycle 4 matches with C_4 = 57.6%, P_4 = 8 kg.

Each cycle follows the recursive state equations P_k = P_{k-1} - 8 and C_k = C_{k-1} - (5 + 0.2 P_{k-1}). Evaluating step by step yields the exact matching pairs for all four cycles.

Adım Adım Çözüm

Calculate state values at the end of Cycle 1 (k = 1)

P_1 = 40 - 8 = 32 kg; C_1 = 100 - (5 + 0.2 * 40) = 100 - 13 = 87.0%

Use the state transition formula using initial values P_0 = 40 and C_0 = 100.

Calculate state values at the end of Cycle 2 (k = 2)

P_2 = 32 - 8 = 24 kg; C_2 = 87.0 - (5 + 0.2 * 32) = 87.0 - 11.4 = 75.6%

Apply the recursive transition rules to the state obtained at k = 1.

Calculate state values at the end of Cycle 3 (k = 3)

P_3 = 24 - 8 = 16 kg; C_3 = 75.6 - (5 + 0.2 * 24) = 75.6 - 9.8 = 65.8%

Apply the recursive transition rules to the state obtained at k = 2.

Calculate state values at the end of Cycle 4 (k = 4)

P_4 = 16 - 8 = 8 kg; C_4 = 65.8 - (5 + 0.2 * 16) = 65.8 - 8.2 = 57.6%

Apply the recursive transition rules to the state obtained at k = 3.

Anahtar Kavram

Sequential Process and Recursive State Transition Tracking

Tahmini Süre:2m 0s

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