The image shows a problem in economic and business mathematics, where a producer's production details over six months are given. Here's a breakdown of the problem:

Problem Summary:

1. Given:

   • In the 3rd month, the production was 28,000 units.
   • The total production over 6 months was 174,000 units.

2. Questions to solve:

   • a) The monthly increase in production.
   • b) The production in the first month.
   • c) Production in the 10th month.
   • d) Total production over 1 year.

Let's assume the production follows an arithmetic progression (AP):

• The production in the nth month can be represented as:
  $P_n = P_1 + (n - 1) \cdot d$
  where $P_n$ is the production in the $n$-th month, $P_1$ is the first month's production, and $d$ is the monthly increase in production.

Step-by-Step Solution Approach:

1. For month 3, we know:
   $P_3 = P_1 + 2d = 28,000$.

2. The total production over 6 months:
   $\text{Total}_6 = \frac{6}{2} \cdot (2P_1 + 5d) = 174,000$.

Now, we can solve these two equations simultaneously to find $P_1$ (first month's production) and $d$ (monthly increase).

Let me calculate this.From the calculations:

• The production in the first month ($P_1$) is 24,000 units.
• The monthly increase in production ($d$) is 2,000 units.

Now let's proceed to answer the remaining questions:

3. Production in the 10th month:
   Using the formula $P_n = P_1 + (n - 1) \cdot d$, we can find $P_{10}$.

4. Total production over 1 year (12 months):
   The total production for 12 months in an arithmetic progression is given by:
   $\text{Total}_\text{12} = \frac{12}{2} \cdot (2P_1 + 11d)$.

I'll calculate these values next.The remaining results are:

• Production in the 10th month: 42,000 units.
• Total production over 1 year (12 months): 420,000 units.

Would you like further details on the steps involved, or do you have any other questions?

Here are some related questions to expand your understanding:

1. What assumptions are made in using an arithmetic progression for production?
2. How would the results change if production followed a geometric progression instead?
3. How could seasonal variations affect monthly production in real-world scenarios?
4. What are other methods to model production growth over time?
5. How can we calculate total production if the growth rate $d$ changes mid-year?

Tip: Always check if the production pattern fits a linear progression; otherwise, explore other models (like exponential growth).