in a set of number rods,there is one rod of length 10,2 rods of length 9,3 of length 8,and so on.If all these rods were laid end to end in one line,how long would the line be ?
To find the total length of the line formed by these number rods, we need to calculate the sum of the lengths of all the rods.
Given information:
1 rod of length 10
2 rods of length 9
3 rods of length 8
and so on.
To find the length of the line, we can start by listing out the length of each rod:
Length of 10: 1 rod
Length of 9: 2 rods
Length of 8: 3 rods
Length of 7: 4 rods
Length of 6: 5 rods
Length of 5: 6 rods
Length of 4: 7 rods
Length of 3: 8 rods
Length of 2: 9 rods
Length of 1: 10 rods
Now, let's calculate the total length:
Total length = (10 * 1) + (9 * 2) + (8 * 3) + (7 * 4) + (6 * 5) + (5 * 6) + (4 * 7) + (3 * 8) + (2 * 9) + (1 * 10)
Simplifying the calculation:
Total length = 10 + 18 + 24 + 28 + 30 + 30 + 28 + 24 + 18 + 10
Total length = 220
Therefore, the length of the line formed by these number rods would be 220 units.
To find out the length of the line formed by laying all the rods end to end, we need to add up the lengths of each individual rod.
Let's start by counting the number of rods of each length:
- Number of rods of length 10: 1
- Number of rods of length 9: 2
- Number of rods of length 8: 3
- Number of rods of length 7: 4
- and so on...
Now, let's calculate the total length of the line:
- Length of 1 rod of length 10 = 10
- Length of 2 rods of length 9 = 2 * 9 = 18
- Length of 3 rods of length 8 = 3 * 8 = 24
- Length of 4 rods of length 7 = 4 * 7 = 28
- and so on...
We can see that the length of the line is increasing by 10 each time. This means that the length of the line formed by laying all the rods end to end can be expressed as an arithmetic series.
Since we want to find the total length, we can use the formula for the sum of an arithmetic series:
Sum = (n/2) * (first term + last term)
In this case, the first term is 10 and the common difference between each term is -1. We need to find the last term.
Let's calculate the last term:
n = Number of rods of each length
Number of roods of length 1 = 10 - (1 - 1) = 10
Number of roods of length 2 = 10 - (1 + 2 - 1) = 9
Number of rods of length 3 = 10 - (1 + 2 + 3 - 1) = 8
Number of rods of length 4 = 10 - (1 + 2 + 3 + 4 - 1) = 7
And so on...
From this pattern, we can see the formula for finding the last term of each length:
Last term = 10 - (1 + 2 + 3 + ... + (n-1))
Now, we can calculate the last term for each length and find the total length:
For rods of length 10:
last term = 10 - (1) = 9
length = (1/2) * (10 + 9) = 9.5
For rods of length 9:
last term = 10 - (1 + 2) = 7
length = (2/2) * (10 + 7) = 17
For rods of length 8:
last term = 10 - (1 + 2 + 3) = 4
length = (3/2) * (10 + 4) = 21
For rods of length 7:
last term = 10 - (1 + 2 + 3 + 4) = 0
length = (4/2) * (10 + 0) = 20
Now, we can add up the lengths of all the rods:
Total length = 9.5 + 17 + 21 + 20 + ...
This is an infinite geometric series, and we can find the sum using the formula:
Sum = (first term) / (1 - common ratio)
In this case, the first term is 9.5, and the common ratio is 10/9 (since the length increases by 10 and each length is 9 less than the previous length).
Sum = 9.5 / (1 - (10/9))
= 9.5 / (9/9 - 10/9)
= 9.5 / (-1/9)
= -9.5 * 9
= -85.5
Therefore, the length of the line formed by laying all the rods end to end is -85.5 units.