SOLUTION: A circular flower bed is boarded by a 3-foot walk. The area of the walk is 7/9 of the area of the bed. What is the radius of the bed?

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Question 896512: A circular flower bed is boarded by a 3-foot walk. The area of the walk is 7/9 of the area of the bed. What is the radius of the bed?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
ONE WAY TO SOLVE THE PROBLEM:
R= radius of the flower bed, in feet.
R%2B3= radius of the whole thing (flower bed plus walkway), also in feet.
pi%2AR%5E2= surface area of the flower bed (in square feet).
pi%2A%28R%2B3%29%5E2= surface area of the whole thing (flower bed plus walkway), also in square feet.
pi%2A%28R%2B3%29%5E2-pi%2AR%5E2= surface area of the walkway, also in square feet.

The problem says that "The area of the walk is 7/9 of the area of the bed," so
pi%2A%28R%2B3%29%5E2-pi%2AR%5E2=%287%2F9%29%2Api%2AR%5E2
That is our equation.
We have to simplify it, and solve it for R .
pi%2A%28R%2B3%29%5E2-pi%2AR%5E2=%287%2F9%29%2Api%2AR%5E2
We expand %28R%2B3%29%5E2 .
pi%2A%28R%5E2%2B6R%2B9%29-pi%2AR%5E2=%287%2F9%29%2Api%2AR%5E2
We take out pi as a common factor.
pi%2A%28R%5E2%2B6R%2B9-R%5E2%29=%287%2F9%29%2Api%2AR%5E2
We simplify (the terms R%5E2 and -R%5E2 cancel each other.
pi%2A%286R%2B9%29=%287%2F9%29%2Api%2AR%5E2
We simplify by dividing both sides of the equal sign by pi .
6R%2B9=%287%2F9%29%2AR%5E2
We multiply both sides of the equal sign times 9 to get rid of the fraction.
9%2A%286R%2B9%29=9%2A%287%2F9%29%2AR%5E2 --> 54R%2B81=7R%5E2
We add -54R-81 to both sides of the equal sign to get all the terms on one side and %220%22 on the other side.
54R%2B81-54R-81=7R%5E2-54R-81 --> 0=7R%5E2-54R-81 <--> 7R%5E2-54R-81=0
We solve using the quadratic formula.
The quadratic formula says that the solutions to ax%5E2%2Bbx%2Bc=0 are given by
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
In the case of 7R%5E2-54R-81=0 ,
a=7 , b=-54 and -81 , so the solutions can be calculated as
x+=+%28-%28-54%29+%2B-+sqrt%28%28-54%29%5E2-4%2A7%2A%28-81%29+%29%29%2F%282%2A7%29+
x=%2854+%2B-+sqrt%282916%2B2268+%29%29%2F14
x=%2854+%2B-+sqrt%285184%29%29%2F14
x=%2854+%2B-+72%29%2F14
x=%2854%2B72%29%2F14=126%2F14=9 is the solution we want.
The radius of the flower bed is highlight%289+feet%29 .
The other solution, x=%2854-72%29%2F14=-18%2F14=-7%2F9 makes no sense, because the measurement of the radius of the flower bed cannot be a negative number.

ANOTHER WAY TO SOLVE THE PROBLEM:
R= radius of the flower bed, in feet.
R%2B3= radius of the whole thing (flower bed plus walkway), also in feet.
You may have been taught that for similar figures, the ratio of areas is the square of the ratio of corresponding lengths.
In the case of circles, the ratio of the areas of two circles is the square of the ratio of their radii.
If you know that, you could use ratios and not even have to know how to calculate the area of a circle, or mention pi .
So the area of the big circle formed by the flower bed plus the walkway is
%28%28R%2B3%29%2FR%29%5E2 times the area of the flower bed,
and the area of the walkway is
%28%28R%2B3%29%2FR%29%5E2-1 times the area of the flower bed.
The problem says that %28%28R%2B3%29%2FR%29%5E2-1=7%2F9 .
%28R%2B3%29%5E2%2FR%5E2-1=7%2F9 --> %28%28R%2B3%29%5E2-R%5E2%29%2FR%5E2=7%2F9 --> 9%2A%28%28R%2B3%29%5E2-R%5E2%29=7%2AR%5E2 --> 9%2A%28R%5E2%2B6R%2B9-R%5E2%29=7%2AR%5E2 --> 9%2A%286R%2B9%29=7%2AR%5E2 --> 54R%2B81%29=7R%5E2 --> 7R%5E2-54R-81=0
That is the same quadratic equation we found above, and could be solved the same way.
You could also solve it by factoring.
For that you need to know factors of 81%2A7=567=9%2A9%2A7=9%2A63 .
It is a big number, and it has 10 different factors paired up into 5 pairs.
You just need a pair of factors whose difference is 54, and 63-9=54 so 63 and 9 are the factors you need.
7R%5E2-54R-81=0
We write that -54R as %28-63%2B9%29R=-63R%2B9R and get
7R%5E2-63R%2B9R-81=0
We group:
%287R%5E2-63R%29%2B%289R-81%29=0
We take out common factors in each group:
7R%28R-9%29%2B9%28R-9%29=0
We take out the common factor %28R-9%29 .
%287R%2B9%29%28R-9%29=0
The factored equation above has the results:
7R%2B9=0 --> R=-9%2F7 and
R-9=0 ---> highlight%28R%2B9%29