SOLUTION: Hello, I have a question that im extremely stuck on.The question is as follows:- A circular pool has a radius r. It is surrounded by a path which is 2 metres wide. Work out a formu
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Question 174119: Hello, I have a question that im extremely stuck on.The question is as follows:- A circular pool has a radius r. It is surrounded by a path which is 2 metres wide. Work out a formula for the area of the path and give its area when r=2.5 metres.Please could you show the step by step guide on how i would work this out.The more simplified the better.
Many thanks Found 2 solutions by ptaylor, ankor@dixie-net.com:Answer by ptaylor(2198) (Show Source):
You can put this solution on YOUR website! Area of Pool=A(pool)=pi(r^2)
Total Area=A(total)=pi(r+2)^2=pi(r)^2+4pi(r)+4pi
(expanding(r+2)^2=r^2+4r+4 and multiplying each term by pi)
Area of path=A(path)=A(total)-A(pool)=pi(r+2)^2-pi(r)^2=
A(path)=pi(r)^2+4pi(r)+4pi-pi(r)^2=
A(path)=4pi(r)+4pi=4pi(r+1) if r=2.5 m, then:
A(path)=4pi*(2.5+1)=4pi*3.5=14pi sq meters;
A(path)=14*3.14=43.95 sq meters
CK
A(total)=pi*(4.5)^2=63.585 sq meters
Area of Pool=pi*(2.5)^2=19.625 sq meters
63.585-19.625=43.96
43.96=43.96
Hope this helps---ptaylor
You can put this solution on YOUR website! A circular pool has a radius r. It is surrounded by a path which is 2 metres wide. Work out a formula for the area of the path:
:
The area (A1) of the pool:
A1 = pi*r^2
;
The area (A2) of the pool and path together:
The radius of this would be the pool radius plus 2 m; (r+2)
:
A2 = pi*(r+2)^2
:
FOIL (r+2)^2
A2 = pi*(r^2 + 4r + 4)
:
Multiply everything inside the brackets by pi
A2 = pir^2 + 4pir + 4pi (overall area)
:
The area of the path (Ap) is the overall area - the pool area: (Ap) = A2 - A1 so we have
:
Ap = (pir^2 + 4pir + 4pi) - pi*r^2
:
Group like terms
Ap = pir^2 - pir^2 + 4pir + 4pi; pir^2's cancel
:
Ap = 4pir + 4pi
:
Factor out 4pi:
Ap = 4pi(r + 1); the formula
;
:
"and give its area when r = 2.5 metres"
:
Substitute 2.5 for r in the above equation
Ap = 4pi(2.5+1)
Ap = 4pi(3.5)
Ap = 43.98 ~ 44 sq meters is the area of the path
:
:
Check to see if this is true (overall radius; 2 + 2.5 = 4.5:
Overall = pi*4.5^2 = 63.62
pool A = pi*2.5^2 = 19.63
-----------------------------
path area -----------44.00; confirms our solution
;
Was this understandable to you? Any questions?
