QUESTION
If the perimeter of square region S and the perimeter of circular region C are equal, then the ratio of the area of S to the area of C is closest to
(A) 3/2
(B) 4/3
(C) 3/4
(D) 2/3
(E) 1/2
SOLUTION
Note it’s a ratio problem, so you’ll be dividing.
Also, you’ll need to make a connection between the length of the square side with the radius of the circle.
Given: perimeter of S = circumference of C
Let length of side of S be L
Perimeter P= 4L
And therefore L=P/4
Let radius of C be r
Circumference c= 2πr
But P=c
So 4L= 2πr
So L = πr/2
Area of square = L² = (πr/2)² = π²r²/4
Area of circle = πr²
Dividing area of S by area of C, we obtain:
(π²r²/4)/ πr²
Cancelling, we obtain:
π/4,
which is approx ¾.
Answer choice C.
it may appear to be a lot of work to do within two minutes. however once you've made the connection between length of side and radius, the working is not too bad.
As always, email me at gmathero@gmail.com if you've got any questions.
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Great solution. Is there a way to start off by plugging in numbers for perimeter S and circumference C?
ReplyDeleteJust got back from a vacation. Sorry for the late reply and lack of posts.
ReplyDeleteYes, you could pick numbers. A good tactic is to be aware of the answer choices when you pick. Note that 3 occurs in four choices, so you could pick it (3) or its square (9), and work backwards.