Six identical machines produce 270 bottles per minute at a constant rate. At the same rate, how many bottles can ten such machines produce in 4 minutes?

Introduction:
When identical machines run at a constant rate, output is directly proportional to both the number of machines and the time for which they run. We can extract a per-machine-per-minute rate and scale up to the requested configuration and duration.

Given Data / Assumptions:

• 6 machines → 270 bottles per minute total.
• All machines identical; rate is linear in machine count and time.
• We need output of 10 machines in 4 minutes.

Concept / Approach:
First find per-machine-per-minute output, then multiply by the number of machines and by the number of minutes. This is a direct unitary-method application without any hidden complications.

Step-by-Step Solution:

Verification / Alternative check:
Proportion method: Output ∝ machines * time → New/Old = (10/6) * (4/1) = 6.666… → 270 * 6.666… = 1800, confirming the calculation.

Why Other Options Are Wrong:
1080 or 648 undercount the scaling; 1700 and 2160 do not match the exact proportional increase from the baseline case.

Common Pitfalls:
Forgetting to multiply by both machine-count and time, or assuming diminishing returns (not stated here). The problem specifies identical machines and constant rate.

Final Answer:
1800