The circumference of a circle is 10pi. Which of the following is not

Question

The circumference of a circle is  10\pi . Which of the following is not a possible value of the area of a rectangle inscribed in it?

A. 30
B. 40
C.  20\sqrt{2} 
D.  30\sqrt{2} 
E.  40\sqrt{2}

TestPrepUnlimited · Answer

There is no limit on how small this rectangle can be, but the biggest rectangle that can fit inside would be the square that is inscribed in the circle. Therefore we can just pick the biggest choice here as there is only 1 option that is incorrect.

Ans: E

If you need the math: The radius of the circle is 5, this corresponds to half of the diagonal of the square inscribed in the circle. Then the square's area is 10*5 = 50. Choice E is the biggest so it is most suspicious,  40*\sqrt{2} > 40*1.4 > 50  so E is out. The next biggest choice is D, which is about 30*1.4 = 42 so it is within 50.

Chasingthesun · Answer

I took a different approach
from the given circumference we can find out the radius and thus the diagonals of rectangle 
diagonals divide the rectangle into 4 isosceles triangles 
each side will be equal to 5(root2) and thus max area will be 50

since option E is beyond 50, it cannot be the area of rectangle inscribed in a circle

pranavmnnofficial · Answer

Radius of the circle=5. Therefore Area is 25pi = 78.5
E) 40 root 2 > 78.5

KarishmaB · Answer

Think about it - what is the constraint on a rectangle inside a fixed circle? It can have very very little area (say the blue rectangle). Then we can start expanding it till it becomes the yellow rectangle (which is actually a square). If we try to expand it any more, its area will start decreasing again till it becomes very very small again (the green rectangle). Screenshot 2021-10-21 at 11.42.34.png Hence, there is no minimum area of the rectangle but there is a maximum possible area dependent on the value of the radius. So if 40 is an acceptable area of a rectangle, then 30 is certainly acceptable too. If 30*sqrt(2) is acceptable (which is about 42), then 40, 30 and 20*sqrt(2) are certainly acceptable too. Hence, the only value that would not be acceptable is the greatest value given i.e. 40*sqrt(2). Answer (E)

adityasuresh · Answer

R = 5, when diagonal is drawn in a rectangle it passes through the centre of the circle circumscribing it. The diagonal divides the rectangle into a 45-45-90 right triangle.The side opposite 90 is the length of the diagonal,so 10=root2X, so x=5root2. This makes the length and breadth of the rectangle the same, area = (5root2)^2=50 this is the maximum area as the area of rectangle is at its maximum when it’s a square,
Option e says area could be 40root2 = which is slight greater than 50(root2=1.4)

The circumference of a circle is 10pi. Which of the following is not

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The circumference of a circle is 10pi. Which of the following is not

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615 to 715 in 15 Days (Ria's Strategies)

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