a piece of wire of length 136(pai) is cut to form 8 circles. the radius of the circles differ from each other, in sequence, by 1 cm.

a) find the radius,r
b) find the number of complete circles that can be formed if the original length of the wire is 190(pai)

How long is a "pai" ? I am not familiar with that unit of length

3.142

I think you mean pi, right?

http://www.google.com/search?rlz=1C1GGGE_enUS379US379&gcx=c&sourceid=chrome&ie=UTF-8&q=pi

yup.. pi.. so i want to know how to solve it

pi is the number but what is the dimension? Is the original wire length 136 pi centimeters?

I agree that you probably meant 136π for the length of the wire

let the first circle have a radius of r
then the others are
r+1, r+2 ,.. , r+7
so the circumferences would be
r(2π) + (r+1)(2π) + ... + (r+7)(2π)
= 2π[ r + r+1 + ... + r+7 ]
= 2π[ 4(r + r+7) , using (n/2)(first + last) as the sum of n terms of an AS
= 2π(8r + 28)
= 136π

2π(8r+28) = 136π
8r + 28 = 68
r = 5

check:
circumferences are
2π(5+6+...+12)
using (n/2)(first + last) as the sum of n terms of an AS
= 2π(4)(5+12) = 136π
length of wire = 136π , perfect!

b)
I assume your circles will start with a radius of 5 and increase by 1
let the number of complete circles be n, where n will have to be a whole number

2π(5 + 6 + .. (n-1) ) = 190π
5+6+... + 5+n-1 = 95

(n/2)(5 + 5+n-1) = 95
n(9+n) = 190
n^2 + 9n - 190 = 0
(n-10)(n+19) = 0
n=10 or a negative

So he will be able to form 10 complete circles.

check:

a) To find the radius, r, of the circles, let's break down the problem step by step:

1. We have a length of wire, which is 136(pai), representing the circumference of all 8 circles combined.
2. The radii of the circles differ from each other, in sequence, by 1 cm. Let's assume the radius of the first circle is r cm.
3. The circumference of a circle is given by the formula: C = 2 * π * r, where C is the circumference and r is the radius.
4. We can calculate the total circumference of all 8 circles using the given wire length: 136(pai) = 8 * 2 * π * r.
5. Rearranging the equation, we get: r = (136(pai)) / (8 * 2 * π).
6. Simplifying further, we find: r = (17(pai)) / (π) = 17 cm.

Therefore, the radius of each circle is 17 cm.

b) To find the number of complete circles that can be formed if the original length of the wire is 190(pai), we can use the same approach:

1. Let's assume the radius of each circle is still r cm.
2. The total circumference of all the circles can be calculated by multiplying the number of circles (Circles) by the circumference of each circle: Total Circumference = Circles * 2 * π * r.
3. We can set up an equation using the given wire length: Total Circumference = 190(pai).
4. Substituting the known values, we have: 190(pai) = Circles * 2 * π * r.
5. Rearranging the equation, we find: Circles = 190(pai) / (2 * π * r).
6. Substituting the value of r from part a) (r = 17 cm), we have: Circles = 190(pai) / (2 * π * 17).
7. Simplifying further, we get: Circles ≈ 5.59.

Therefore, the number of complete circles that can be formed with a wire length of 190(pai) is approximately 5.59. Since we cannot have a fraction of a circle, the maximum number of complete circles that can be formed is 5.