The cost of manufacturing 1000 pens in Rs.17,000 and that of manufacturing 1,500 pens is 22,000. If the relationship between the number of units produced and its cost is linear, find the relation and also find the cost of manufacturing 2,000 pens

Given that the cost of manufacturing pens has a linear relationship with the number of units produced, we can represent this relationship with the equation of a straight line:

\[ C = mN + b \]

where:
- \( C \) is the cost of manufacturing,
- \( N \) is the number of units produced,
- \( m \) is the slope of the line (the cost per additional unit),
- \( b \) is the y-intercept (the fixed cost).

We are given the following information:
1. The cost of manufacturing 1000 pens is Rs. 17,000.
2. The cost of manufacturing 1,500 pens is Rs. 22,000.

We can use these points to find \( m \) and \( b \).

First, let's set up two equations based on the given points:

For 1000 pens:
\[ 17000 = 1000m + b \tag{1} \]

For 1500 pens:
\[ 22000 = 1500m + b \tag{2} \]

We solve these equations simultaneously to find \( m \) and \( b \).

Subtract equation (1) from equation (2):
\[ 22000 - 17000 = 1500m + b - (1000m + b) \]
\[ 5000 = 500m \]
\[ m = \frac{5000}{500} \]
\[ m = 10 \]

Now, substitute \( m = 10 \) back into equation (1):
\[ 17000 = 1000(10) + b \]
\[ 17000 = 10000 + b \]
\[ b = 17000 - 10000 \]
\[ b = 7000 \]

So, the linear relationship between the number of units produced \( N \) and the cost \( C \) is:
\[ C = 10N + 7000 \]

To find the cost of manufacturing 2000 pens, we substitute \( N = 2000 \) into the equation:
\[ C = 10(2000) + 7000 \]
\[ C = 20000 + 7000 \]
\[ C = 27000 \]

Therefore, the cost of manufacturing 2000 pens is Rs. 27,000.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:
1. How would the cost change if the fixed cost \( b \) was different?
2. What happens to the cost if the slope \( m \) increases or decreases?
3. Can we determine the cost efficiency between different production levels using this model?
4. How does this linear model help in predicting future manufacturing costs?
5. What are some real-world factors that might cause deviations from this linear model?
6. How can we use this model to calculate the break-even point?
7. What would be the cost of producing zero pens according to this model?
8. How can this model be adjusted if there are bulk discounts for materials?

**Tip:** When solving for the coefficients of a linear equation using two points, always ensure to solve simultaneously to accurately determine both the slope and the intercept.

Learn how to calculate manufacturing costs using linear equations for different production levels. Solve for the cost of producing 2000 pens given costs for 1000 and 1500 pens. Includes step-by-step solution and real-world applications.

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