Widget Production Problem

Setting Up the Problem

Given information:

• Starting widgets: 5 (at 8 AM)
• Rate doubles every 3 hours
• Time period: 8 AM → 8 PM = 12 hours

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Step 1: Identify the Production Periods

Every 3 hours represents one doubling period.

Period	Time Window	Widgets Produced
1	8 AM – 11 AM	5
2	11 AM – 2 PM	5 × 2¹ = 10
3	2 PM – 5 PM	5 × 2² = 20
4	5 PM – 8 PM	5 × 2³ = 40

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Step 2: Calculate Total Production

$$\text{Total} = 5 + 10 + 20 + 40$$

$$\text{Total} = \boxed{75 \text{ widgets}}$$

Using geometric series formula as verification: $$S = a \cdot \frac{r^n - 1}{r - 1} = 5 \cdot \frac{2^4 - 1}{2 - 1} = 5 \times 15 = 75 ✓$$

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Step 3: Calculate Average Hourly Rate

$$\text{Average Rate} = \frac{\text{Total Widgets}}{\text{Total Hours}} = \frac{75}{12}$$

$$\boxed{\approx 6.25 \text{ widgets per hour}}$$

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Summary

Metric	Value
Total widgets produced	75 widgets
Average production rate	6.25 widgets/hour

💡 Key insight: Notice the exponential nature means the last period alone (40 widgets) accounts for more than half of total production — illustrating why doubling growth accelerates so dramatically over time.

To solve this problem, we need to break it down into the time duration, the growth of the widget count, and the calculation of the production rate.

Step 1: Determine the total time duration

The factory starts at 8 AM and ends at 8 PM.

• From 8 AM to 8 PM is a total of 12 hours.

Step 2: Determine the number of doubling periods

The problem states that the rate (or quantity, based on standard exponential growth riddles) doubles every 3 hours.

• Total Time = 12 hours
• Doubling Period = 3 hours
• Number of Periods ($n$) = $12 \div 3 = 4$ periods.

Step 3: Calculate the widget count at each stage

The factory starts with 5 widgets. We apply the doubling effect for each 3-hour period.

• Start (8 AM): 5 widgets
• Period 1 (11 AM): $5 \times 2 = 10$ widgets
• Period 2 (2 PM): $10 \times 2 = 20$ widgets
• Period 3 (5 PM): $20 \times 2 = 40$ widgets
• Period 4 (8 PM): $40 \times 2 = 80$ widgets

By 8 PM, the factory has a total inventory of 80 widgets.

Step 4: Calculate total widgets "produced"

The question asks how many widgets were produced in total. We must distinguish between the final inventory and the new widgets created during the shift.

• Final Inventory: 80 widgets
• Starting Inventory: 5 widgets
• Net Production: $80 - 5 = 75$ widgets

(Note: In some casual contexts, "produced" might refer to the final total count of 80, but strictly speaking, the factory produced 75 new widgets during the 12-hour shift.)

Step 5: Calculate the average production rate

The average production rate is the total amount of new widgets produced divided by the total time taken.

• Total Produced: 75 widgets
• Total Time: 12 hours
• Average Rate: $75 \div 12$

$$75 \div 12 = 6.25$$

Answer: By 8 PM, the factory will have a total of 80 widgets (having produced 75 new widgets). The average production rate is 6.25 widgets per hour.