GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsHard
A scientist is observing the decay of a radioactive substance. The graph shows the amount of substance remaining over time. The graph passes through the points (0, 100) and (2, 25). Which of the following functions best models the amount of substance, A, remaining after t hours?
- AA(t) = 100(0.5)^t
- BA(t) = 100(0.25)^t
- CA(t) = 100 - 37.5t
- DA(t) = 100 - 75t
Show answer & explanationAnswer & explanation
Correct answer: A. A(t) = 100(0.5)^t
The graph passes through (0, 100), which is the y-intercept, meaning the initial amount (a) is 100. This eliminates options A and D as linear decay. Now we have A(t) = 100(b)^t for exponential decay. Using the second point (2, 25): 25 = 100(b)^2. Divide by 100: 0.25 = b^2. Take the square root: b = 0.5. Thus, the function is A(t) = 100(0.5)^t.
Why the other options are wrong
- B. If b=0.25, then A(2) = 100(0.25)^2 = 100(0.0625) = 6.25, which is incorrect.
- C. This is a linear decay model. At t=2, A(2) = 100 - 37.5(2) = 100 - 75 = 25. While it matches the points, radioactive decay is exponential, not linear. Also, the question asks for the 'best' model, and exponential is characteristic for decay.
- D. This is a linear decay model. At t=2, A(2) = 100 - 75(2) = 100 - 150 = -50, which is incorrect.
Modeling Exponential Decay
Exponential decay models describe quantities that decrease at a rate proportional to their current value. The general form is A(t) = a(b)^t, where 'a' is the initial amount, 'b' is the decay factor (0 < b < 1), and 't' is time.
- The decay factor 'b' is found by dividing consecutive terms or by solving for 'b' using two points.
- The initial amount 'a' is the value of A(t) when t=0.
- Exponential decay curves approach the x-axis but never touch it (asymptote).
Memory trick: Start with 'a', multiply by 'b' every 't'!