Math Problem Statement
Solution
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:
- be the production in the first month,
- be the monthly increase in production,
- be the production in the -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:
- be the production in the first month,
- be the monthly increase in production,
- be the production in the -th month.
Known
- (production in the 3rd month),
- The total production over 6 months: .
We can solve for the monthly increase () and the first month's production () 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:
- How would the solution change if the monthly production followed a geometric sequence?
- What would the total production be if the monthly increase doubled?
- How can the production rate be optimized over 12 months to reach a higher total?
- What is the impact of starting with a lower initial production on total annual output?
- 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.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequence
Linear Algebra
Production Increase
Formulas
Arithmetic sequence formula: a_n = a_1 + (n-1) * d
Sum of arithmetic sequence: S_n = n/2 * (a_1 + a_n)
Theorems
Arithmetic progression properties
Suitable Grade Level
Grades 10-12 (High School)
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