SOLUTION: A landscape architect is planning a circular flower bed that will be surrounded by a ring of paving stones, 25 cm wide. The area of the ring of paving stones will be half the ar

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: A landscape architect is planning a circular flower bed that will be surrounded by a ring of paving stones, 25 cm wide. The area of the ring of paving stones will be half the ar      Log On


   

Question 1177833: A landscape architect is planning a circular flower
bed that will be surrounded by a ring of paving
stones, 25 cm wide. The area of the ring of paving
stones will be half the area of the flower bed.
Determine the radius of the flower bed, to the
nearest centimetre.
Found 3 solutions by mananth, greenestamps, ikleyn:
Answer by mananth(16949) About Me  (Show Source):
You can put this solution on YOUR website!

pi*(r+25)^2 -pi*r^2= (1/2) pi*r^2

pi*(r+25)^2 = (3/2) pi*r^2
(r+25)^2= 3/2 * r^2
r^2+50t +625 = 3/2 * r^2
r^2-(3/2)r^2=-625
-(1/2) r^2 =-625
r^2= 1250
r = 35 cm

Answer by greenestamps(13292) About Me  (Show Source):
You can put this solution on YOUR website!


The other tutor got lost in the middle of her calculations....

The area of the ring of paving stones is half the area of the garden, so the area of the garden plus paving stones is 1.5 times the area of the garden.

Radius of the garden = r
Radius of garden plus paving stones = r+25

%28r%2B25%29%5E2+=+%283%2F2%29%28r%5E2%29
%28%28r%2B25%29%2Fr%29%5E2+=+3%2F2
%28r%2B25%29%2Fr+=+sqrt%283%2F2%29+=+1.2247
r%2B25+=+1.2247r
25+=+0.2247r
r+=+25%2F0.2247+=+111 with rounding error

ANSWER: the radius of the flower bed is 111cm.

CHECK: (round numbers)
pi(111^2) = 38708
pi(136^2) = 58106
58106/38708 = 1.5

Answer by ikleyn(53560) About Me  (Show Source):
You can put this solution on YOUR website!
.
A landscape architect is planning a circular flower bed that will be surrounded by a ring of paving
stones, 25 cm wide. The area of the ring of paving stones will be half the area of the flower bed.
Determine the radius of the flower bed, to the nearest centimetre.
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        The solution by @mananth is incorrect due to arithmetic errors on the way.
        I came to bring a correct solution. 


pi*(r+25)^2 - pi*r^2 = (1/2)*pi*r^2


Cancel pi in both sides


(r+25)^2 - r^2 = (1/2)*r^2


r^2 + 50r + 625 - r^2 = (1/2)*r^2


50r + 625 = (1/2)r^2


r^2 - 100r - 1250 = 0


Use the quadratic formula and find  r = 50+%2B-+25%2Asqrt%286%29.


It gives a unique meaningful positive solution x = 50+%2B+25%2Asqrt%286%29 = 111 cm, rounded.    ANSWER

Solved.