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which of the following is a possible value for y?
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03 Nov 2021, 07:19
03 Nov 2021, 07:19
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Question Stats:
70% (02:39) correct
30% (02:18) wrong based on 20 sessions
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If \(8 < x < 9\), and \(x^2 = (10 – y)(10 + y)\), which of the following is a possible value for y?
(A) –7
(B) –6
(C) 3
(D) 4
(E) 5
(A) –7
(B) –6
(C) 3
(D) 4
(E) 5
which of the following is a possible value for y?
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Updated on: 03 Nov 2021, 08:56
Updated on: 03 Nov 2021, 08:56
Using the property, \((a+b)(a-b) = a^2 - b^2\)
\(x^2 = 10^2 - y^2\)
\(x^2 + y^2 = 10^2\)
We are given that 8 < \(x\) < 9
Lets create constraint for y based on constraints of \(x\)
\(x > 8\), then \(y^2 < 100 - 64 = 36\) , therefore \(y < 6\)
\(x < 9\), then \(y^2 > 100 - 81 = 19\) , therefore \(y > 4.35\)
Therefore, \(4.35 < y < 6\)
Only one option satisfies the above constraint which is 5.
Hence, the answer is E.
\(x^2 = 10^2 - y^2\)
\(x^2 + y^2 = 10^2\)
We are given that 8 < \(x\) < 9
Lets create constraint for y based on constraints of \(x\)
\(x > 8\), then \(y^2 < 100 - 64 = 36\) , therefore \(y < 6\)
\(x < 9\), then \(y^2 > 100 - 81 = 19\) , therefore \(y > 4.35\)
Therefore, \(4.35 < y < 6\)
Only one option satisfies the above constraint which is 5.
Hence, the answer is E.
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Re: which of the following is a possible value for y?
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03 Nov 2021, 08:46
03 Nov 2021, 08:46
1
Carcass wrote:
If \(8 < x < 9\), and \(x^2 = (10 – y)(10 + y)\), which of the following is a possible value for y?
(A) –7
(B) –6
(C) 3
(D) 4
(E) 5
(A) –7
(B) –6
(C) 3
(D) 4
(E) 5
If \(8 < x < 9\), then \(8^2 < x^2 < 9^2\).
Simplify to get: \(64 < x^2 < 81\)
Since \(x^2 = (10 – y)(10 + y)\), we can substitute this value into the above inequality to get: \(64 < (10 – y)(10 + y) < 81\)
Expand to get: \(64 < 100 – y^2 < 81\)
Subtract \(100\) from all sides to get: \(-36 < –y^2 < -19\)
Multiply all sides by \(-1\) to get: \(36 > y^2 > 19\) [since we multiplied all sides of the inequality by a NEGATIVE value, we REVERSED the direction of the inequality symbols]
At this point, we can easily see that \(y = 5\) satisfies the inequality \(36 > y^2 > 19\)
Answer: E
