**Q) ****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 6 ^{th} year and 1130 rollers in the 9^{th} 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) (a) What was the company’s production in the 8 ^{th} year? **

**OR**

**(iii) (b) What was the company’s total production in the first 6 years?**

**Ans:**

**STEP BY STEP SOLUTION**

In this question, productions are being increased every year at same amount, hence it forms an AP..

We know that the nth term of an AP is given by, T_{n } = a + (n – 1) d

Where, a = first term of the AP

d = common difference of the AP

Hence, 6^{th} term, T_{6} = a + ( 6 – 1) d

In the given question, we are given T_{6} = 800

∴ 800 = a + 5 d …………………. (i)

Hence, 9^{th} term, T_{9} = a + ( 9 – 1) d

In the given question, we are given T_{9} = 1130

∴ 1130 = a + 8 d …………………(ii)

By deducting equation (i) from equation (ii), we get:

(a + 8 d) – (a + 5 d) = 1130 – 800

∴ 3 d = 330

∴ d = 110

**(i) First Year production:**

By substituting value of d in equation (i), we get:

a + 5 (110) = 800

∴ a = 800 – 550 = 250

**Company’s production in first year is 250.**

**(ii) Annual increase in Company’s production:**

We just calculated above, common difference d = 110

**Hence, the increase in company’s production every year is 110 **

**(iii) (a) Production in 8 ^{th} year:**

Production of 8^{th} year will be 8^{th} term of the AP

Value of 8^{th} term is given by, T_{8} = a + (8 – 1) d

By substituting values of a = 250 and d = 110, we get:

∴ T_{8} = 250 + (8 – 1) (110) = 250 + 770 = 1020

**Hence, 1,020 road rollers will be produced in 8 ^{th} year.**

**(iii) (b) Total production in first 6 years:**

The total production in multiple years is given by sum of the terms

We know that the sum of n terms is given by,

**Therefore, total 3,150 road rollers will be produced in 6 years.**

**Please do press “Heart” button if you liked the solution.**