Hello mastermin346 Originally Posted by

**mastermin346** A roll of thread with 90pi cm long is cut into 5 parts to make 5 circles.The radii of the circles increase by 1 cm consecutively.Calculate

a)the radius of the smallest circle

b)the circumference of the 6 th circle

c)the number of circles obtained if the original length of the thread is 660pi cm

A number of points:

1 There's no reason to suppose there was a typo in the original question: the length of the thread could quite well be $\displaystyle 90\pi$ cm; in which case, the radius of the smallest circle is $\displaystyle 7$ cm.

2 This is an example of an arithmetic progression, not a geometric progression.

3 For part (c), if the initial radius is $\displaystyle 7$ cm and there are $\displaystyle n$ circles altogether, then the total length of thread used is:$\displaystyle 2\pi\big(7+(7+1) + (7+2) + ... + (7+[n-1])\big)$$\displaystyle =2\pi\times \tfrac12n(13+n)$, using the AP formula $\displaystyle S = \tfrac12n(a+l)$

$\displaystyle =660\pi$

This gives a quadratic equation for $\displaystyle n$ with one positive root.

Grandad