GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsHard
A scientist is observing the decay of a radioactive isotope. The amount of the isotope remaining, A(t), in grams, after t days is given by the function A(t) = 100 * (0.5)^(t/3). What does the value '0.5' represent in this function?
- AThe initial amount of the isotope.
- BThe decay rate per day.
- CThe half-life of the isotope.
- DThe fraction of the isotope remaining after each half-life period.
Show answer & explanationAnswer & explanation
Correct answer: D. The fraction of the isotope remaining after each half-life period.
In an exponential decay function of the form A(t) = A_0 * b^(t/k), A_0 is the initial amount, 'b' is the decay factor (often 0.5 for half-life), and 'k' is the half-life period. Here, 0.5 specifically represents the fraction remaining after one half-life period has passed.
Why the other options are wrong
- A. The initial amount is 100 grams, not 0.5.
- B. The decay rate per day would be 1 - (0.5)^(1/3), not just 0.5.
- C. The half-life is '3' days (the denominator of the exponent), not 0.5.
Exponential Decay Function Components
In A(t) = A_0 * b^(t/k), A_0 is initial amount, b is decay factor (e.g., 0.5 for half-life), and k is the decay period (e.g., half-life).
- A_0 = initial amount
- b = decay factor (fraction remaining after one period)
- k = length of one decay period
- t = time elapsed
Memory trick: Start with A_0, Decay by B, Time by K!