The image shows a mathematics problem related to business and economics, focusing on production. Here’s the problem statement and questions translated:

Economics and Business Mathematics Problem

A producer manufactures 28,000 units of goods in the third month, while the total production over six months amounts to 174,000 units.

Determine:

• a) The increase in production each month.
• b) The production amount in the first month.
• c) The production amount in the tenth month.
• d) The total production over one year.

Let’s assume the monthly production follows an arithmetic sequence, where the production increases by a constant amount each month.

Let:

• $a_1$ be the production in the first month,
• $d$ be the monthly increase in production,
• $a_n = a_1 + (n-1)d$ be the production in the $n$-th month.

Known dataThe image shows a mathematics problem related to business and economics, focusing on production. Here’s the problem statement and questions translated:

Economics and Business Mathematics Problem

A producer manufactures 28,000 units of goods in the third month, while the total production over six months amounts to 174,000 units.

Determine:

• a) The increase in production each month.
• b) The production amount in the first month.
• c) The production amount in the tenth month.
• d) The total production over one year.

Let’s assume the monthly production follows an arithmetic sequence, where the production increases by a constant amount each month.

Let:

• $a_1$ be the production in the first month,
• $d$ be the monthly increase in production,
• $a_n = a_1 + (n-1)d$ be the production in the $n$-th month.

Known

• $a_3 = 28,000$ (production in the 3rd month),
• The total production over 6 months: $S_6 = 174,000$.

We can solve for the monthly increase ($d$) and the first month's production ($a_1$) using this information.

I will calculate the answers now.The solution to the problem is as follows:

• The production in the first month (a₁) is 24,000 units.
• The monthly increase in production (d) is 2,000 units.

Now, let's calculate the other parts:

• The production in the 10th month,

• The total production over 1 year (12 months). I will compute these now.Here are the final answers:

• The production in the 10th month is 42,000 units.

• The total production over one year (12 months) is 420,000 units.

Do you want further details or explanations on any of these steps?

Here are 5 related questions you might want to explore:

1. How would the solution change if the monthly production followed a geometric sequence?
2. What would the total production be if the monthly increase doubled?
3. How can the production rate be optimized over 12 months to reach a higher total?
4. What is the impact of starting with a lower initial production on total annual output?
5. How could a change in demand affect the production increase over time?

Tip: Arithmetic progressions are often used to model gradual increases in production or cost, making it a powerful tool for planning and forecasting in business.