HELP-PROBLEM SOLVING IN MATHEMATICS
posted by alya on .
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
I think you mean pi, right?
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)
2π(8r+28) = 136π
8r + 28 = 68
r = 5
using (n/2)(first + last) as the sum of n terms of an AS
= 2π(4)(5+12) = 136π
length of wire = 136π , perfect!
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.