A road roller is a compactor-type engineering vehicle, used to compact soil, gravel, concrete, etc, in the construction of roads and foundations. They are also used at landfills or in agriculture. A company started making road rollers 10 years ago and increased its production uniformly by a fixed number every year. The company produces 800 rollers in the 6th year and 1130 rollers in the 9th year.

 

A road roller is a compactor-type engineering vehicle, used to compact soil, gravel, concrete, etc, in the construction of roads and foundations. They are also used at landfills or in agriculture. A company started making road rollers 10 years ago and increased its production uniformly by a fixed number every year. The company produces 800 rollers in the 6th year and 1130 rollers in the 9th year.

Based on the above information, answer the following questions :

(i) What is the company’s production in the first year ?

(ii) What was the increase in the company’s production every year ?

(iii) What was the company’s production in the 8^th year ?

(iv) What was the company’s total production in the first 6 years ?

Solution:

We can solve this problem with Arithmetic Sequence.

Let, ‘a’ be the production in the first year

And

        ‘d’ be the annual increase in in the production

Given that

        Production in 6^th year = 800 units.

        We know that a_n = a + (n-1)d

Substitute the values, we get

        a + (6-1)d = 800

i.e.,   a + 5d = 800

(or)   a + 5d – 800      (1)

This is the first system of equation.

According to the next condition,

Production in the 9^th year = 1130

i.e.,   a + (9 – 1)d = 1130

i.e.,   a + 8d = 1130

(or)  a + 8d – 1130            (2)

This is the second system of equations

We have to solve these system of equations

        a + 5d – 800 and

        a + 8d – 1130

Subtract the first equation from the second equation

We get,

        (a + 8d) – (a + 5d ) = 1130 - 800

        3d  = 330

        d  = 110

Substitute the value of d back into the first equation

We get,

        a + 5 x 110 = 800

        a + 550 = 800

        a = 800 – 550 = 250.

(i)                  Hence the production in the first year  a = 250 units

(ii)                 The increase in the company’s production in every year
d= 110

(iii)                Company’s production in the 8^th year:\

We know that,

        a = 250 and

        d = 110

So,    a_n = a + (n – 1) d

        a₈ = 250 + (8 – 1) x 110

            = 250 + 7 x 110

            = 250 + 770

           =  1020 units

Hence the company’s production in the 8^th year = 1020 units

iv)  The company’s total production in the first 6 years

        Here we need to find the sum of production,

        We know that,

                Sum = (n/2) x (first term + last term)

                First term = a = 250

                Last term = a₆ = a + 5d = 250 + 5 x 110 = 800

                n = 6

                Sum = (6/2) x (250 + 800)

                        =  3 x 1050

                        = 3150 units

The Final Answer is:

(i)                  The company’s production in the first year is 250 rollers.

(ii)                 The increase in the company’s production every year is 110 rollers.

(iii)                The company’s production in the 8^th year is 1020 rollers

(iv)               The company’s total production in the first 6 years in 3150 rollers.