A radioactive material decays at a rate of change proportional to the current amount, Q, of the radioactive material. Which equation describes this relationship? Choose 1 answer: (A) Q(t)=-Q^(kt) (B) Q(t)=-kQ (C) (dQ)/(dt)=-kQ (D) (dQ)/(dt)=-Q^(kt)

Question

A radioactive material decays at a rate of change proportional to the current amount,  Q, of the radioactive material. Which equation describes this relationship? Choose 1 answer: (A)  Q(t)=-Q^(kt) (B)  Q(t)=-kQ (C)  (dQ)/(dt)=-kQ (D)  (dQ)/(dt)=-Q^(kt)

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Problem Statement: The problem states that the rate of change of the radioactive material is proportional to the current amount of the material. This means that if we let Q represent the quantity of the radioactive material, then the rate of change of Q with respect to time t, denoted as (dQ)/(dt), is proportional to Q itself. The constant of proportionality is typically represented by a negative constant -k, because the material is decaying. We need to find the equation that correctly represents this relationship.Option (A): Option (A) Q(t) = -Q^(kt) suggests that the quantity Q at time t is equal to the negative of Q raised to the power of kt. This does not represent a rate of change and does not make sense in the context of decay, which is typically exponential and not a power function of the current amount.Option (B): Option (B) Q(t) = -kQ suggests that the quantity Q at time t is equal to the negative constant times the current quantity. This is not a rate of change but rather a static equation that does not involve time differentiation.Option (C): Option (C) (dQ)/(dt) = -kQ correctly represents the rate of change of the quantity Q with respect to time t as being proportional to the current amount Q. The negative sign indicates that the quantity is decreasing over time, which is consistent with decay. This matches the description given in the problem.Option (D: Option (D) (dQ)/(dt) = -Q^(kt) suggests that the rate of change of Q with respect to time is equal to the negative of Q raised to the power of kt. This does not represent a simple proportional relationship and is not the standard form for exponential decay.

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