To solve this problem, we need to model the total production after $t$ hours, considering the production increase over time. Here's how we can approach this:

Step 1: Understand the Problem

• In the first hour, the factory produces 100 units.
• In each subsequent hour, production increases by 5 units.

Thus, production per hour forms an arithmetic sequence:

• Production in the 1st hour: $100$
• Production in the 2nd hour: $100 + 5 = 105$
• Production in the 3rd hour: $100 + 2 \times 5 = 110$
• And so on.

To calculate the total production after $t$ hours, we sum up the first $t$ terms of this sequence.

Step 2: Formula for Total Production

The sum of the first $t$ terms of an arithmetic sequence is given by: $S_t = \frac{t}{2} \cdot (a + l)$ where:

• $t$ is the number of terms (hours),
• $a = 100$ is the first term,
• $l = 100 + (t-1) \cdot 5$ is the last term.

Substituting $l$: $S_t = \frac{t}{2} \cdot \left(100 + \left[100 + (t-1) \cdot 5\right]\right)$

Simplify: $S_t = \frac{t}{2} \cdot \left(200 + 5(t-1)\right)$ $S_t = \frac{t}{2} \cdot (200 + 5t - 5)$ $S_t = \frac{t}{2} \cdot (195 + 5t)$ $S_t = \frac{5t^2 + 195t}{2}$

Thus, the total production after $t$ hours is: $P(t) = \frac{5t^2 + 195t}{2}$

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Step 3: Calculate Production After 8 Hours

Substitute $t = 8$ into the function $P(t)$: $P(8) = \frac{5(8)^2 + 195(8)}{2}$ $P(8) = \frac{5(64) + 195(8)}{2}$ $P(8) = \frac{320 + 1560}{2}$ $P(8) = \frac{1880}{2} = 940$

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Final Answer

The total production after 8 hours is: $\boxed{940 \text{ units}}$

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Do you want a step-by-step breakdown of how the arithmetic sequence formula was derived or clarification on any part?

5 Related Questions

1. How does the formula for an arithmetic sequence's sum apply to other problems in production or economics?
2. How would the function change if production increased by a percentage rather than a fixed number of units?
3. What is the average production rate over the 8-hour period?
4. How can this model be adapted for more complex production patterns, like exponential growth?
5. If production stops after 12 hours, how many total units would be produced?

Tip: Always identify if the pattern of growth is arithmetic, geometric, or something else before selecting the formula!