AAPL wrote: ↑Tue Jan 28, 2020 4:29 am
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 lengths of two legs are a and b cm, respectively; thus, the hypotenuse = √(a^2 + b^2) cm
Area of the mirror = ab/2
We have to determine whether ab/2 < 10 or ab < 20.
Let's take each statement one by one.
1) The perimeter of the mirror is \(24 \,cm\).
=> 2(a + b) = 24 => a + b = 12
Case 1: Say a = 1; thus, b = 11. Thus, ab = 1*11 = 11 < 20. The answer is yes.
Case 2: Say a = 6; thus, b = 6. Thus, ab = 6*6 = 36 > 20. The answer is no.
No unique answer. Insufficient.
2)The diagonal of the mirror is less than \(11 \,cm\).
=> √(a^2 + b^2) < 11
a^2 + b^2 < 121
Case 1: Say a = 2; thus, b = 10. Thus, (a^2 + b^2) = (2^2 + 10^2) = 104 < 121. Thus, ab = 2*10 = 20 = 20. The answer is no.
Case 2: Say a = 6; thus, b = 6. Thus, (a^2 + b^2) = (6^2 + 6^2) = 72 < 121. Thus, ab = 6*6 = 36 > 20. The answer is no.
No unique answer. Insufficient.
(1) and (2) together
We know that a + b = 12 => (a + b)^2 = 144 => a^2 + b^2 + 2ab = 144 => 2ab = 144 – (a^2 + b^2)
Again, we know that a^2 + b^2 < 121, from 2ab = 144 – (a^2 + b^2), we can write that 2ab > 144 – 121
Or, 2ab > 23 => ab > 11.5. The answer is no. Sufficient.
The correct answer:
C
Hope this helps!
-Jay
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