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A radioactive material decays at a rate of change proportional to the current amount, , of the radioactive material.Which equation describes this relationship?Choose answer:(A) (B) (C) (D)
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Q. A radioactive material decays at a rate of change proportional to the current amount, , of the radioactive material.Which equation describes this relationship?Choose answer:(A) (B) (C) (D)
- 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 represent the quantity of the radioactive material, then the rate of change of with respect to time , denoted as , is proportional to itself. This can be expressed mathematically as , where is the proportionality constant and the negative sign indicates that the quantity is decreasing over time.
- Mathematical Expression: Now let's examine the given options to identify which one correctly represents this relationship:(A) (B) (C) (D) Option (A) directly matches our mathematical expression of the rate of change being proportional to the current amount, with a negative sign indicating decay.Option (B) suggests that the quantity at time is equal to a negative power function, which does not represent a rate of change and also does not make physical sense because the quantity cannot be negative.Option (C) suggests that the rate of change of is proportional to a power function of , which is not the simple proportionality described in the problem.Option (D) suggests that the quantity at time is directly proportional to the negative of the proportionality constant times , which does not represent a rate of change and is also incorrect.
