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21 Dec 2023, 06:02
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E
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60% (02:04) correct
40%
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wrong
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The rate R at which a chemical reaction in a certain industrial process proceeds is a function of time t and is given by \(R = at^3 + bt^2 + c\), where a, b, and c are constants and t > 0. Is there a positive value of t for which R = 0 ?
(1) a > b
(2) c > 0
(1) a > b
(2) c > 0
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21 Dec 2023, 07:06
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The rate R at which a chemical reaction in a certain industrial process proceeds is a function of time t and is given by \(R = at^3 + bt^2 + c\), where a, b, and c are constants and t > 0. Is there a positive value of t for which R = 0 ?
If we leave all that mambo jumbo out, the question essentially asks whether \(at^3 + bt^2 + c = 0\), has a solution for t, where t > 0.
(1) a > b
If all three constants are positive and t is positive, then \(at^3 + bt^2 + c\) becomes the sum of three positive values and cannot equal 0. So, in this case, no value of positive t would satisfy \(at^3 + bt^2 + c = 0\).
However, in other cases, for example, if a = 0, b = -1, and c = 1, the equation becomes t^2 = 1, which has a positive t satisfying it (t = 1).
Not sufficient.
(2) c > 0.
Similarly, we can deduce the insufficiency of this statement.
(1)+(2) Combining the statements, we still lack sufficient information. Again, we can consider the cases:
If all three constants are positive and t is positive, then \(at^3 + bt^2 + c\) becomes the sum of three positive values and cannot equal 0. So, in this case, no value of positive t would satisfy \(at^3 + bt^2 + c = 0\).
However, in other cases, for example, if a = 0, b = -1, and c = 1, the equation becomes t^2 = 1, which has a positive t satisfying it (t = 1).
Not sufficient.
Answer: E.
If we leave all that mambo jumbo out, the question essentially asks whether \(at^3 + bt^2 + c = 0\), has a solution for t, where t > 0.
(1) a > b
If all three constants are positive and t is positive, then \(at^3 + bt^2 + c\) becomes the sum of three positive values and cannot equal 0. So, in this case, no value of positive t would satisfy \(at^3 + bt^2 + c = 0\).
However, in other cases, for example, if a = 0, b = -1, and c = 1, the equation becomes t^2 = 1, which has a positive t satisfying it (t = 1).
Not sufficient.
(2) c > 0.
Similarly, we can deduce the insufficiency of this statement.
(1)+(2) Combining the statements, we still lack sufficient information. Again, we can consider the cases:
If all three constants are positive and t is positive, then \(at^3 + bt^2 + c\) becomes the sum of three positive values and cannot equal 0. So, in this case, no value of positive t would satisfy \(at^3 + bt^2 + c = 0\).
However, in other cases, for example, if a = 0, b = -1, and c = 1, the equation becomes t^2 = 1, which has a positive t satisfying it (t = 1).
Not sufficient.
Answer: E.
30 Jun 2024, 14:50
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turd_fergsn
No, it does not mean that. If that were the case, then why would we even need the statements? We could say right away that yes, for some specific values of a, b, and c, there is a positive value of t for which R equals 0. However, that's not what the question means. We are given an expression R = at^3 + bt^2 + c, where a, b, and c represent some specific unknown numbers and two additional pieces of information (a > b and c > 0). The question asks whether we have sufficient information to say for sure that there is a positive value of t for which R equals 0, which, as explained above, we don't.
