Calculating Printers Needed for Specific Output
This problem involves calculating the number of printers required based on the rate of printing, the number of pages, and the time available. We can solve this using a proportional reasoning approach.
Work-Rate Formula Application
The relationship between the number of printers (P), the time spent (T), and the work done (W) can be expressed as:
\( \frac{P_1 \times T_1}{W_1} = \frac{P_2 \times T_2}{W_2} \)
Where:
- \(P_1\) = Initial number of printers
- \(T_1\) = Initial time
- \(W_1\) = Initial work (pages printed)
- \(P_2\) = Required number of printers
- \(T_2\) = Required time
- \(W_2\) = Required work (pages to be printed)
Step-by-Step Calculation
- Identify Given Values:
- \(P_1 = 5\) printers
- \(T_1 = 6\) hours
- \(W_1 = 9,000\) pages
- \(T_2 = 8\) hours
- \(W_2 = 28,800\) pages
- \(P_2 = ?\)
- Apply the Formula:
Substitute the known values into the formula:
\( \frac{5 \times 6}{9000} = \frac{P_2 \times 8}{28800} \)
- Simplify the Equation:
Calculate the left side:
\( \frac{30}{9000} = \frac{1}{300} \)
The equation becomes:
\( \frac{1}{300} = \frac{P_2 \times 8}{28800} \)
- Solve for \(P_2\):
Rearrange the equation to isolate \(P_2\):
\( P_2 = \frac{1}{300} \times \frac{28800}{8} \)
\( P_2 = \frac{1}{300} \times 3600 \)
\( P_2 = \frac{3600}{300} \)
\( P_2 = 12 \)
Conclusion
Therefore, 12 printers are needed to print 28,800 pages in 8 hours.
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