Question
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)
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)
Answers
How long is a "pai" ? I am not familiar with that unit of length
3.142
Writeacher
I think you mean <i>pi</i>, right?
http://www.google.com/search?rlz=1C1GGGE_enUS379US379&gcx=c&sourceid=chrome&ie=UTF-8&q=pi
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:
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:
Related Questions
A piece of wire 100 cm long is cut into two pieces. One piece is bent to form a square and the other...
A piece of wire is wound exactly 100 times around a circular tube with a radius of 9.8 centimeters....
A piece of wire is wound exactly 100 times, without overlap, around a circular
tube with a radius...
A 100 inch piece of wire is cut into two pieces. Each piece of wire is used to make a square wire fr...