Digital SATMath: AlgebraMedium
A scientist is observing the decay of a radioactive substance. The amount of the substance, A, in grams, remaining after t days is given by the equation A = 250(0.8)^t. What is the approximate half-life of this substance, in days?
- A2.8 days
- B3.1 days
- C1.5 days
- D3.8 days
Show answer & explanationAnswer & explanation
Correct answer: B. 3.1 days
The half-life is the time it takes for the substance to decay to half of its initial amount. Initial amount is 250g, so half is 125g. We need to solve 125 = 250(0.8)^t. Dividing by 250 gives 0.5 = (0.8)^t. Taking the natural logarithm of both sides: ln(0.5) = t * ln(0.8). So, t = ln(0.5) / ln(0.8) ≈ -0.693 / -0.223 ≈ 3.107 days.
Why the other options are wrong
- A. This value might be a result of setting 0.8^t = 0.5 and solving incorrectly.
- C. This value might arise from incorrect logarithmic calculations or estimation.
- D. This value might be a result of using an incorrect base for the logarithm or other calculation errors.
Half-Life in Exponential Decay
The half-life of a substance undergoing exponential decay is the time required for half of the initial quantity to decay.
- Set the final amount to half of the initial amount.
- Solve the exponential equation using logarithms.
- Applies to radioactive decay, drug clearance, etc.
Memory trick: Half-life: Halve the start, then log the time!