Twelve iron poles are to be used for pillars and the lengths of the pole form an A.P if the 3rd Pole is 4m and the 7th pole is 10m.Find the
A): Length of the pole in order of the A.P.
B): Cost of all the poles if a metre cost #45.
3rd Pole is 4m ----> a + 2d = 4
the 7th pole is 10m --> a + 6d = 10
subtract those two equations:
4d = 6
d = 6/4 = 3/2
back in a + 2d = 4
a + 2(3/2) = 4
a + 3 = 4
a = 1
Pole(n) = 3 + (n-1)(3/2)
= 1 + (3/2)n - 3/2
= -1/2 + 3/2 n
= 1/2 (3n - 1)
for the cost, you need the sum of the length of 12 poles
that is,
1 + 5/2 + 4 + 11/2 + 7 + .... + 35/2
= 6(first + last_
= 6(1 + 35/2) = 111 metres is your total length
each metre costs 45, so
just do 45*111
A): To find the length of the poles in the arithmetic progression (A.P.), we need to determine the common difference first.
Given:
3rd pole = 4m
7th pole = 10m
The number of terms between the 3rd pole and the 7th pole is (7 - 3) - 1 = 4 - 1 = 3.
Therefore, the common difference (d) is the difference in length divided by the number of terms:
d = (10 - 4) / 3 = 6 / 3 = 2
Now that we have the common difference, we can find the lengths of the poles in the A.P.:
1st pole = 3rd pole - 2d = 4 - 2(2) = 4 - 4 = 0m
2nd pole = 1st pole + d = 0 + 2 = 2m
3rd pole = 4m
4th pole = 3rd pole + d = 4 + 2 = 6m
5th pole = 4th pole + d = 6 + 2 = 8m
6th pole = 5th pole + d = 8 + 2 = 10m
7th pole = 10m
So, the lengths of the poles in the A.P. are:
0m, 2m, 4m, 6m, 8m, 10m
B): To calculate the cost of all the poles, we need to find the sum of their lengths and multiply it by the cost per meter.
Sum of lengths = 0m + 2m + 4m + 6m + 8m + 10m = 30m
Cost per meter = #45
Total cost = Cost per meter × Sum of lengths = #45 × 30 = #1350
Therefore, the cost of all the poles is #1350.
A) To find the length of the poles in the order of the arithmetic progression (AP), we need to determine the common difference (d).
We have two known terms of the AP: the 3rd pole, which is 4m (a3 = 4), and the 7th pole, which is 10m (a7 = 10).
We can use the formula for the nth term of an AP to find the common difference:
an = a1 + (n - 1)d
For the 3rd pole:
a3 = a1 + (3 - 1)d
4 = a1 + 2d
For the 7th pole:
a7 = a1 + (7 - 1)d
10 = a1 + 6d
We now have a system of two equations involving a1 and d:
4 = a1 + 2d ----(1)
10 = a1 + 6d ----(2)
To solve for a1 and d, we can subtract equation (1) from equation (2):
10 - 4 = (a1 + 6d) - (a1 + 2d)
6 = 4d
Divide both sides by 4:
d = 6/4 = 3/2 = 1.5
Now that we have the common difference (d = 1.5), we can find the values of the poles by substituting back into equation (1):
4 = a1 + 2(1.5)
4 = a1 + 3
a1 = 4 - 3
a1 = 1
Therefore, the length of the poles in the order of the AP is:
a1 = 1m, a2 = 1 + 1.5 = 2.5m, a3 = 1 + 2(1.5) = 4m, a4 = 1 + 3(1.5) = 5.5m, and so on.
B) To find the cost of all the poles, we need to know the total length of the poles first.
The sum of an AP can be calculated using the formula:
Sn = (n/2)(2a1 + (n - 1)d)
In this case, n = 12 (twelve poles) and a1 = 1m (first pole).
Sn = (12/2)(2(1) + (12 - 1)(1.5))
Sn = 6(2 + 11(1.5))
Sn = 6(2 + 16.5)
Sn = 6(18.5)
Sn = 111m
The total length of the poles is 111m.
To find the cost of all the poles, we multiply the total length by the cost per meter:
Cost = Total length × Cost per meter
Cost = 111m × #45/m
Cost = #4995
Therefore, the cost of all the poles is #4995.
To find the length of the poles in the given arithmetic progression (A.P.), we need to determine the common difference (d) first.
The third pole is 4m (a3 = 4) and the seventh pole is 10m (a7 = 10). We can use these two values to find the common difference.
The formula to find the nth term (an) in an arithmetic progression (A.P.) is given by:
an = a1 + (n - 1)d
Here, a1 is the first term, n is the term number, and d is the common difference.
We can use the formula to find the common difference (d) by substituting the values of a3 and a7:
a7 = a1 + (7 - 1)d
10 = a1 + 6d
a3 = a1 + (3 - 1)d
4 = a1 + 2d
We now have a system of equations:
a1 + 6d = 10 ...(1)
a1 + 2d = 4 ...(2)
Solving equations (1) and (2) simultaneously will give us the values of a1 and d.
Subtracting equation (2) from equation (1):
(a1 + 6d) - (a1 + 2d) = 10 - 4
4d = 6
d = 6/4
d = 3/2
Substituting the value of d in equation (2):
a1 + 2(3/2) = 4
a1 + 3 = 4
a1 = 4 - 3
a1 = 1
Therefore, the first term (a1) in the A.P. is 1, and the common difference (d) is 3/2.
To find the length of the poles, we can use the nth term formula:
an = a1 + (n - 1)d
For the length of each pole, let's use a1 as the first term, d as the common difference, and n as the term number.
A) Length of the pole in order of the A.P.:
The first term (a1) = 1
The common difference (d) = 3/2
Using the nth term formula, we can find the length of each pole:
a1 = 1 + (1 - 1)(3/2) = 1 meter
a2 = 1 + (2 - 1)(3/2) = 4/2 = 2 meters
a3 = 1 + (3 - 1)(3/2) = 7/2 = 3.5 meters
a4 = 1 + (4 - 1)(3/2) = 10/2 = 5 meters
a5 = 1 + (5 - 1)(3/2) = 13/2 = 6.5 meters
a6 = 1 + (6 - 1)(3/2) = 16/2 = 8 meters
a7 = 1 + (7 - 1)(3/2) = 19/2 = 9.5 meters
a8 = 1 + (8 - 1)(3/2) = 22/2 = 11 meters
a9 = 1 + (9 - 1)(3/2) = 25/2 = 12.5 meters
a10 = 1 + (10 - 1)(3/2) = 28/2 = 14 meters
a11 = 1 + (11 - 1)(3/2) = 31/2 = 15.5 meters
a12 = 1 + (12 - 1)(3/2) = 34/2 = 17 meters
Therefore, the lengths of the poles in the given arithmetic progression are as follows: 1m, 2m, 3.5m, 5m, 6.5m, 8m, 9.5m, 11m, 12.5m, 14m, 15.5m, 17m.
B) Cost of all the poles:
To find the cost of all the poles, we need to multiply the length of each pole by the cost per meter, which is #45.
The total cost (C) is calculated by summing up the individual costs of each pole:
C = (a1 + a2 + a3 + ... + a12) * cost per meter
Substituting the lengths of each pole into the formula, we have:
C = (1 + 2 + 3.5 + 5 + 6.5 + 8 + 9.5 + 11 + 12.5 + 14 + 15.5 + 17) * 45
Calculating the sum of the lengths:
C = 135 * 45
C = 6075
Therefore, the cost of all the poles is #6075.