Geometric progression

**mastermin346** · January 14th 2010, 08:13 PM

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

**pickslides** · January 14th 2010, 10:18 PM

Hi there 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

Well i'm reading this as the length of string is 90cm?

Therefore

90 = length of string #1 + length of string #2+ .... +length of string #5

And the difference in each length, shaped as a circle is, adding 1 to the next radius so

$90 = 2\pi r+2\pi (r+1)+2\pi (r+2)+2\pi (r+3)+2\pi (r+4)$

Factoring out and then dividing $2\pi$

$90 = 2\pi (r+ (r+1)+ (r+2)+ (r+3)+ (r+4))$

$\frac{90}{2\pi} = (r+ (r+1)+ (r+2)+ (r+3)+ (r+4))$

Grouping like terms on the RHS

$\frac{90}{2\pi} = 5r+10$

$\frac{45}{\pi} = 5r+10$

$\frac{45}{\pi} -10= 5r$

$r= \frac{\frac{45}{\pi} -10}{5}$

Originally Posted by mastermin346

b)the circumference of the 6 th circle
c)the number of circles obtained if the original length of the thread is 660pi cm

You should be able to use the same logic to finish the rest.

**Grandad** · January 15th 2010, 06:24 AM

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 $90\pi$ cm; in which case, the radius of the smallest circle is $7$ cm.

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

3 For part (c), if the initial radius is $7$ cm and there are $n$ circles altogether, then the total length of thread used is: