Question 830276
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Hi, there--

THE PROBLEM:
There are two circular gardens A and B.The circumference of garden A is 1.760Km and the 
area of garden B is 25 times the area of garden A. Find the circumference of garden B.

A SOLUTION:

I can think of several ways to solve this problem: a long way and a short way. Here's the 
LONG WAY first…

We know the circumference of Garden A. We can work backwards to find the area of garden A.
The formula for the circumference C of a circle with radius r is
{{{C=2*pi*r}}}

Substitute 1.760 for C and solve for r.
{{{1.760=2*pi*r}}}
{{{r=1.760/(2*pi)}}}
{{{r=0.880/pi}}}

The radius of garden A is {{{0.880/pi}}} km. We won't give an approximation for {{{pi}}} until 
the final step of the problem.)

Given the radius, we can work backwards to find the area of garden A. The formula for the
area of a circle with radius r is
{{{A=pi*r^2}}}

Substitute {{{0.880/pi}}} for r and solve for S.
{{{A=pi*(0.880/pi)^2}}}
{{{A=pi*(0.7744/(pi)^2)}}}
{{{A=0.7744/pi}}}

The area of garden B is 25 times the area of garden A. To find the area of garden B, multiply
the area of garden A by 25. 

{{{(0.7744/pi)*25=19.36/pi}}}

The area of Garden B is {{{19.36/pi}}} square km. We can work backwards to find the radius. 
Substitute {{{19.36/pi}}} for A and solve for r.

{{{A=pi*r^2}}}
{{{19.36/pi=pi*r^2}}}
{{{r^2=19.36/(pi)^2}}}
{{{r =4.4/pi}}}

Given the radius of Garden B, we can determine its circumference. Substitute {{{4.4/pi}}} for r 
and simplify.

{{{C=2*pi*r}}}
{{{C=2*pi*(4.4/pi)}}}
{{{C=8.8}}}

The circumference of Garden B is 8.8 km.

THE SHORTER WAY:
Because the gardens are both circles, we know that they are similar figures.
The circumferences of the two circles will vary by some factor, and the areas will vary by the square of that factor. 

We are given that the areas of the circles vary by a factor of 25. Therefore, the circumferences
will vary by a factor of {{{sqrt(25)=5}}}. In other words, the circumference of B is 5 times the 
circumference of A.

The circumference of garden B is (5)(1.760)=8.8 km. The LONG WAY is actually a proof that
this SHORT WAY works. Never do more work than you have to. Let the math do the work for 
you!!!

Hope this helps! Feel free to email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com
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