Question

$\huge\underline\mathcal{\red{Case \ study}}$

Lumber is a significant natural resource that contributes jobs to the US economy. Lumber companies source their raw materials from privately-managed or government-leased forests. In order to process tree wood into usable lumber, this raw material is transported to lumber mills, where it is cut to different sizes. Lumber is primarily used by the construction industry, though it can also be used to produce furniture, paper and pulp, and composites such as plywood. A lumber company stacks 200 logs in the manner as shown in the attachment.

20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on.

$\implies$ Based on the above information answer the following questions:

1.1) No. of logs in first row, second row, third row,....
(a) Follow a pattern forming an A.P. with common difference 1.
(b) Follow a pattern forming an A.P. with common difference -1.
(c) Don't follow any specific pattern.
(d) follow a pattern forming an A.P. with common difference 2.

1.2) The no. of rows in which 200 logs are stacked is
(a) 25
(b) 20
(c) 16
(d) 10

1.3) The no. of logs in the top row is
(a) 5
(b) 7
(c) 10
(d) 2

1.4) The no. of logs in the middle rows are:
(a) 11, 10
(b) 12, 11
(c) 14, 13
(d) 13, 12

1.5) The no. of logs in the top two rows is
(a)10
(b)11
(c)9
(d)12

__________________________________
Note:-
$\Rightarrow$ The above case study is based on the chapter-5 (Arithmetic progression) of class 10.

$\bullet{\leadsto}$ Quality answers required!

Kindly:-
$\bullet{\leadsto}$ Give proper explanation.
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Answer 1

Answers!!

(1.1) No. of logs in first row, second row, third row,...

(b) follow a pattern forming an A.P. with common difference -1.

Reason:- The first term is 20 and the second term is 19. So, the common difference (d) = 19 - 20 = -1

(1.2) The no. of rows in which 200 logs are stacked is

(c) 16

Reason:- a = 20, d = -1

$\sf S_{n}=200$

$\sf S_{n}=\dfrac{n}{2}(2a+(n-1)d)$

$\sf 200=\dfrac{n}{2}(2(20)+(n-1)(-1))$

$\sf 400=n(40+1-n)$

400 = 41n - n²

n² - 41 + 400 = 0

n² - 25n - 16n + 400 = 0

(n - 25)(n - 16) = 0

n = 25

or n = 16

$\sf T_{25}=a+24d=20-24=-4$

$\sf T_{16}=a+15d=20-15=5$

Rejecting the negative term, we get the answer.

(1.3) The no. of logs in the top row is

(a) 5

Reason:- $\sf T_{16}=a+15d=20-15=5$

(1.4) The no. of logs in the middle rows are

(d) 13, 12

Reason:- 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5

Number of observations = 16

Median = ((n/2)th term + ((n/2) + 1)th term) ÷ 2

Median = ((16/2)th term + ((16/2) + 1)th term) ÷ 2

Median = (8th term + 9th term) ÷ 2

The 8th and the 9th term is 13 and 12, respectively. The rows cannot be in points so we won't solve the median further.

(1.5) The no. of logs in the top two rows is

(b) 11

Reason:- The bottom rows are 20, 19,....6 and 5. So, the logs in the top two rows are 6 + 5 = 11.

Answer 2

Answer:

1)b

2)c

3)a

4)d

5)b

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