A certain radioactive substance decays exponentially with yearly decay factor of 0.93. There are initially 107 grams present.
The Correct Answer and Explanation is :
The decay of a radioactive substance follows the exponential decay formula:
[
A(t) = A_0 \times (decay \ factor)^t
]
where:
- ( A(t) ) is the amount remaining after time ( t ),
- ( A_0 ) is the initial amount,
- ( decay \ factor ) is 0.93,
- ( t ) is the number of years.
Correct Answer:
The equation describing the decay of the given substance is:
[
A(t) = 107 \times (0.93)^t
]
This formula allows us to calculate the remaining amount of the substance after any given number of years.
Explanation:
Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. This is commonly observed in radioactive substances, where atoms disintegrate over time, losing mass at a predictable rate.
In this case, the decay factor is 0.93, meaning that every year, 93% of the substance remains, and 7% is lost. This makes it different from linear decay, where a fixed amount disappears each year.
Let’s break it down:
- Initial State: At ( t = 0 ), the substance starts with 107 grams.
- After 1 Year: The mass will be ( 107 \times 0.93 = 99.51 ) grams.
- After 2 Years: The mass will be ( 107 \times 0.93^2 = 92.54 ) grams.
- Long-term Behavior: Over many years, the mass will approach zero but never fully disappear.
This exponential decay model is widely used in physics, chemistry, and even finance (e.g., depreciation of assets). Understanding how substances decay helps in nuclear physics, carbon dating, and medical imaging.
Now, I’ll generate a graph illustrating the decay.
Here is the graph illustrating the exponential decay of the radioactive substance over time. The curve shows how the amount decreases gradually, following the equation ( A(t) = 107 \times (0.93)^t ). Let me know if you need further clarification!