In an exponential decay model, the exponent counts how many intervals of change have passed. When the question changes the time unit, convert the length of one interval to the new unit, then divide the new time variable by that length. Change the exponent's time unit, not the starting amount or the fraction remaining. Multiplying by the conversion factor instead reverses the time scale.
Hints
- Hint 1
The exponent counts intervals: when it equals , the mass has been multiplied by once. What value of makes ?
- Hint 2
Each hour has minutes, and each minute has seconds. Convert the whole six-hour interval to seconds before changing the equation.
- Hint 3
The new exponent must count how many halving intervals fit into seconds. Divide elapsed seconds by the seconds in one interval; keep the starting mass and unchanged.
Step-by-step
Convert the halving interval
Step 1Identify when the mass halves
Let be the mass after hours. Type and then . Desmos shows , half of the starting . So the exponent counts six-hour halving intervals: at , it equals .
- Step 2
Convert one interval to seconds
Convert the six-hour interval, not the mass. Each hour has minutes, and each minute has seconds. Type ; Desmos gives seconds per halving.
- Step 3
Make the exponent count seconds-based intervals
Divide elapsed seconds by seconds per halving interval, so the new exponent is . Don't multiply by : that would count more halvings as the interval gets longer. The starting mass and halving factor stay unchanged, giving milligrams after seconds. Choice D.