A radioactive material decays at a rate of change proportional to the current amount, $Q$ , of the radioactive material. $\newline$ Which equation describes this relationship? $\newline$ Choose $1$ answer: $\newline$ (A) $Q(t)=-Q^{k t}$ $\newline$ (B) $Q(t)=-k Q$ $\newline$ (C) $\frac{d Q}{d t}=-k Q$ $\newline$ (D) $\frac{d Q}{d t}=-Q^{k t}$

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Q. A radioactive material decays at a rate of change proportional to the current amount,

Q

, of the radioactive material.

\newline

Which equation describes this relationship?

\newline

Choose

1

answer:

\newline

(A)

Q(t)=-Q^{k t}

\newline

(B)

Q(t)=-k Q

\newline

(C)

\frac{d Q}{d t}=-k Q

\newline

(D)

\frac{d Q}{d t}=-Q^{k t}

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 $\frac{dQ}{dt}$ , 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) $\frac{dQ}{dt} = -kQ$ 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) $\frac{dQ}{dt} = -Q^{kt}$ 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.