Question

What is the company’s production in the first year ?

Answer 1

The production forms an arithmetic progression (AP) where:
\[a = \text{Production in the first year}, \quad d = \text{Increase in production every year}.\]
The production in the $n$-th year is given by:
\[a_n = a + (n-1)d.\]
For the 6th year:
\[800 = a + (6 - 1)d \implies 800 = a + 5d. \tag{1}\]
For the 9th year:
\[1130 = a + (9 - 1)d \implies 1130 = a + 8d. \tag{2}\]
Subtract (1) from (2):
\[1130 - 800 = a + 8d - (a + 5d) \implies 330 = 3d \implies d = 110.\]
Substitute $d = 110$  into (1):
\[800 = a + 5(110) \implies 800 = a + 550 \implies a = 250.\]
Correct Answer: Production in the first year = 250 rollers.

Answer 2

Step 1: Define the variables.
Let the number of road rollers produced in the \(n\)-th year be denoted by \(P_n\). We are told that the production increases uniformly by a fixed number every year. So, the production in the \(n\)-th year can be expressed as:
$$ P_n = P_1 + (n-1) \times d $$
Where: - \(P_1\) is the production in the first year. - \(d\) is the fixed number by which production increases each year.
Step 2: Use the given data.
From the problem, we know the following: - In the 6th year, the company produces 800 rollers, so \(P_6 = 800\). - In the 9th year, the company produces 1130 rollers, so \(P_9 = 1130\).
Using the formula for \(P_n\), we can write two equations: 1. \( P_6 = P_1 + (6-1) \times d = 800 \), so \( P_1 + 5d = 800 \). 2. \( P_9 = P_1 + (9-1) \times d = 1130 \), so \( P_1 + 8d = 1130 \).
Step 3: Solve the system of equations.
We now have the following system of equations: 1. \( P_1 + 5d = 800 \) 2. \( P_1 + 8d = 1130 \) To solve for \(d\), subtract the first equation from the second: $$ (P_1 + 8d) - (P_1 + 5d) = 1130 - 800 $$ $$ 3d = 330 $$ $$ d = \frac{330}{3} = 110 $$
Step 4: Conclusion.
The increase in the company’s production every year is \(d = 110\) rollers.

Answer 3

(a) Step 1: Use the formula for the number of road rollers produced in the \(n\)-th year.
We know that the production in the \(n\)-th year follows the formula:
$$ P_n = P_1 + (n-1) \times d $$
Where: - \(P_1\) is the production in the first year. - \(d\) is the fixed increase in production every year.
From the previous calculation, we know: - The increase in production every year is \(d = 110\). - In the 6th year, the production was 800 rollers, so \(P_6 = 800\).
Step 2: Find \(P_1\) (production in the first year).
Using the equation for the 6th year:
$$ P_6 = P_1 + 5d = 800 $$
Substitute \(d = 110\):
$$ P_1 + 5 \times 110 = 800 $$
$$ P_1 + 550 = 800 $$
$$ P_1 = 800 - 550 = 250 $$
So, the production in the first year, \(P_1 = 250\) rollers.
Step 3: Calculate the production in the 8th year.
Now that we know \(P_1 = 250\) and \(d = 110\), we can calculate the production in the 8th year using the formula:
$$ P_8 = P_1 + (8-1) \times d $$
Substitute the values of \(P_1\) and \(d\):
$$ P_8 = 250 + 7 \times 110 $$
$$ P_8 = 250 + 770 = 1020 $$
Step 4: Conclusion.
The company’s production in the 8th year was \(P_8 = 1020\) rollers.


(b) Step 1: Formula for total production.
The total production in the first \(n\) years can be found by summing up the productions in each year. The production in the \(n\)-th year follows the formula:
$$ P_n = P_1 + (n-1) \times d $$
Where: - \(P_1\) is the production in the first year. - \(d\) is the fixed increase in production every year.
We know: - The increase in production every year is \(d = 110\). - The production in the first year is \(P_1 = 250\).
Step 2: Calculate the production in each of the first 6 years.
Using the formula for \(P_n\), we can calculate the production for each of the first 6 years: 1. \(P_1 = 250\) (given). 2. \(P_2 = P_1 + (2-1) \times d = 250 + 1 \times 110 = 250 + 110 = 360\). 3. \(P_3 = P_1 + (3-1) \times d = 250 + 2 \times 110 = 250 + 220 = 470\). 4. \(P_4 = P_1 + (4-1) \times d = 250 + 3 \times 110 = 250 + 330 = 580\). 5. \(P_5 = P_1 + (5-1) \times d = 250 + 4 \times 110 = 250 + 440 = 690\). 6. \(P_6 = P_1 + (6-1) \times d = 250 + 5 \times 110 = 250 + 550 = 800\).
Step 3: Calculate the total production in the first 6 years.
Now, sum the production from each year:
$$ \text{Total Production} = P_1 + P_2 + P_3 + P_4 + P_5 + P_6 $$
$$ \text{Total Production} = 250 + 360 + 470 + 580 + 690 + 800 $$
$$ \text{Total Production} = 3150 $$
Step 4: Conclusion.
The company’s total production in the first 6 years was 3150 rollers.