The direct relationship between the number of men, the amount of work (tunnel length), and time (days) is proportional, assuming other factors are constant. This can be represented as: \[ \text{Men} \times \text{Work} \propto \text{Time} \]
| Men | Tunnel Length | Days | |
|---|---|---|---|
| Initial | 140 | 1.5 km | 60 |
| Remaining | X | 4.5 km | 140 |
Using the direct proportion, we calculate the number of men (X) required for the remaining work:
\[ X = 140 \times \frac{4.5}{1.5} \times \frac{60}{140} \]
Simplifying the equation:
\[ X = 140 \times 3 \times \frac{60}{140} = 3 \times 60 = 180 \]
Therefore, the total number of men required for the remaining work is \( \boxed{180} \).
The number of additional men needed is the difference between the total required and the initial number of men:
\[ 180 - 140 = \boxed{40} \]
40 additional men are necessary to complete the remaining 4.5 km tunnel within 140 days.
A box contains 16 red, 12 white, and 15 yellow identical marbles. A man picks one marble at a time without replacement. How many times must he pick a marble to be 100% certain of picking at least 3 white marbles?