AMC 10 Weekly Practice Round 2

Complete problem set with solutions and individual problem pages

Problem 23 Medium

The research vessel Xuelong carried out a scientific expedition to Antarctica. Departing from Shanghai at its maximum speed of 19 knots (1 knot = 1 nautical mile per hour), it would take more than 30 days to reach Antarctica. This time, the vessel departed from Shanghai at a speed of 16 knots and, after a certain number of days, successfully arrived at its destination. It then worked in the polar region for a certain number of days before returning at a speed of 12 knots. On the 83rd day after leaving Shanghai, due to weather conditions, its sailing speed dropped to 2 knots. Two days later, it continued at a speed of 14 knots for another 4 days to return to Shanghai. How many days did the Xuelong work in Antarctica?

  • A.

    1

  • B.

    2

  • C.

    3

  • D.

    4

  • E.

    5

Answer:C

Let x be the number of days the Xuelong took to travel from Shanghai to Antarctica, and y be the number of days it worked in Antarctica.

Then: 16x = (82 - x - y) \times 12 + 2 \times 2 + 14 \times 4

Simplifying gives the indeterminate equation: 7x + 3y = 261

From this, we find: y = 10 \quad \text{or} \quad y = 3

When y = 10, x = 33; when y = 3, x = 36.

Since traveling from Shanghai to Antarctica at the maximum speed of 19 knots (1 knot = 1 nautical mile/hour) takes more than 30 days, we have: 16x \geqslant 19 \times 30 \quad \Rightarrow \quad x \geqslant \frac{19 \times 30}{16} = 35\frac{5}{8}

Therefore, y = 3 fits the conditions of the problem.

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