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, T₆ = a + ( 6 – 1) d

In the given question, we are given T₆  = 800

∴ 800 = a + 5 d …………………. (i)

Hence, 9^th term, T₉  = a + ( 9 – 1) d

In the given question, we are given T₉  = 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, T₈ = a + (8 – 1) d

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

∴ T₈ = 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.

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