Data Sufficiency

AAPL wrote:Manhattan Prep

Does the rectangular mirror have an area greater than \(10\,cm^2\)?

1. The perimeter of the mirror is \(24\,cm\).
2. The diagonal of the mirror is less than \(11\,cm\).

OA C

Say the length and the breadth of the rectangular mirror are a and b, respectively.

Thus, we have to determine whether ab > 10.

Let's take each statement one by one.

1. The perimeter of the mirror is \(24\,cm\).

=> 2(a + b) = 24
=> a + b = 12

Can't determine if ab > 10. If a = 1 and b = 11, we have ab = 1*11 = 11 > 10, the answer is yes; however, If a = 1/2 and b = 11.5, we have ab = 1/2*11.5 = 5.75 < 10, the answer is no. No unique answer. Insufficient.

2. The diagonal of the mirror is less than \(11\,cm\).

=> a^2 + b^2 < 11^2. Can't determine if ab > 10. If a = 2 and b = 9, we have a^2 + b^2 = 2^2 + 9^2 = 85 < 11^2 and ab = 2*9 = 18 > 10. The answer is Yes; however, if a = b = 2, we have a^2 + b^2 = 2^2 + 2^2 = 4 + 4 = 8 < 11^2, and ab = 2*2 = 4 < 10. The answer is No. No unique answer. Insufficient.

(1) and (2) together

From (1), we have a + b = 12; thus, (a + b )^2 = 144 => a^2 + b^2 + 2ab = 144.
From (2), we have a^2 + b^2 < 11^2 = 121.

Thus, from a^2 + b^2 + 2ab = 144 and a^2 + b^2 < 121, we have 2ab > 144 - 121 => ab > 11.5 > 10. The asnwer is yes. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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