khaos3787
is this one simply try and error ? or is there a specific method ?
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.