SOLUTION: 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.

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Question 830276: 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.
Found 2 solutions by math-vortex, KMST:
Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
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%2Api%2Ar

Substitute 1.760 for C and solve for r.
1.760=2%2Api%2Ar
r=1.760%2F%282%2Api%29
r=0.880%2Fpi

The radius of garden A is 0.880%2Fpi 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%2Ar%5E2

Substitute 0.880%2Fpi for r and solve for S.
A=pi%2A%280.880%2Fpi%29%5E2
A=pi%2A%280.7744%2F%28pi%29%5E2%29
A=0.7744%2Fpi

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. 

%280.7744%2Fpi%29%2A25=19.36%2Fpi

The area of Garden B is 19.36%2Fpi square km. We can work backwards to find the radius. 
Substitute 19.36%2Fpi for A and solve for r.

A=pi%2Ar%5E2
19.36%2Fpi=pi%2Ar%5E2
r%5E2=19.36%2F%28pi%29%5E2
r+=4.4%2Fpi

Given the radius of Garden B, we can determine its circumference. Substitute 4.4%2Fpi for r 
and simplify.

C=2%2Api%2Ar
C=2%2Api%2A%284.4%2Fpi%29
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%2825%29=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

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The two gardens are similar figures, so the ratio of areas is the square of the ratio of any linear measurement, such as the circumference or the radius.
If the ratio of areas is 25=5%5E2, the ratio of the circumferences is 5, and the circumference of garden B is 5 times the circumference of garden A, or
5%2A%281.760Km%29=highlight%288.800Km%29

ANOTHER WAY:
r= radius of the small circle (garden A)
a=pi%2Ar%5E2= area of the small circle (garden A)
R= radius of the large circle (garden B)
A=pi%2AR%5E2= area of the large circle (garden B)
The problem says that A%2Fa=25
So, pi%2AR%5E2%2F%28pi%2Ar%5E2%29=25
R%5E2%2Fr%5E2=25
%28R%2Fr%29%5E2=25
%28R%2Fr%29%5E2=5%5E2
R%2Fr=5 --> R=5r
The circumference of the small circle (garden A) is 2pi%2Ar=1.760Km .
At this point we could solve for r , and then calculate R and then the circumference of garden B.
An easier way forward, is found think in a way a bit more abstract:
The circumference of the large circle (garden B) is
2pi%2AR=2pi%2A%285r%29=5%282pi%2Ar%29=5%2A%281.760Km%29=highlight%288.800Km%29