Question

In a shovel dumper combination, the loading time of a dumper is 3 minutes. The shovel serves 8 dumpers. If the cycle time of a dumper, including its loading time, is 22 minutes, the waiting time of a dumper, in minutes, is:

Hint:

In a shovel-dumper combination, the waiting time is the total cycle time minus the individual dumper's cycle time. Make sure to account for both loading and travel times when calculating cycle times.

Answer 1

Given:

Loading time per dumper (\( L \)) = 3 minutes
Number of dumpers (\( N \)) = 8
Cycle time of a dumper (\( C \)) = 22 minutes (includes loading time)

Objective:

Find the waiting time (\( W \)) of a dumper before it is loaded again.

Approach:

Calculate the total time the shovel takes to load all dumpers once.
Determine when each dumper returns after completing its cycle.
Compute the time difference between when a dumper returns and when the shovel is available to load it again.

Step 1: Total Loading Time for All Dumpers

The shovel loads dumpers sequentially, taking \( L = 3 \) minutes per dumper. For \( N = 8 \) dumpers:
\[ \text{Total loading time} = N \times L = 8 \times 3 = 24\ \text{minutes} \]

Step 2: Dumper Cycle Time

The cycle time (\( C = 22 \) minutes) includes:
\[ C = \text{Loading time} + \text{Travel, dumping, and return time} \] \[ 22 = 3 + \text{Other activities time} \] \[ \text{Other activities time} = 22 - 3 = 19\ \text{minutes} \]

Step 3: Dumper Return and Waiting Time

Each dumper returns after completing its cycle. The shovel takes 24 minutes to cycle through all dumpers.

For any dumper:
\[ \text{Time when loading starts} = t \] \[ \text{Time when dumper returns} = t + 22 \] \[ \text{Time when shovel is free to load it again} = t + 24 \] \[ \text{Waiting time} = (t + 24) - (t + 22) = 2\ \text{minutes} \]

Verification:

Dumper 1:

Starts loading at \( t = 0 \), finishes at \( t = 3 \).
Returns at \( t = 22 \).
Shovel finishes loading all dumpers at \( t = 24 \).
Waits from \( t = 22 \) to \( t = 24 \): \( W = 2 \) minutes.

Dumper 2:

Starts loading at \( t = 3 \), finishes at \( t = 6 \).
Returns at \( t = 25 \).
Shovel loads Dumper 1 at \( t = 24 \), finishes at \( t = 27 \).
Waits from \( t = 25 \) to \( t = 27 \): \( W = 2 \) minutes.

This pattern holds for all dumpers.

General Formula:

The waiting time can also be derived as:
\[ W = \text{Total loading time} - \text{Cycle time} = 24 - 22 = 2\ \text{minutes} \]

Final Answer:

The waiting time of a dumper is \(\boxed{2}\) minutes.