The quantity of a radioactive substance decays according to the function Q (t) = 100e -1/4. where t represents time in years
The given function represents the decay of a radioactive substance over time. The equation Q(t) = 100e^(-1/4t) represents the quantity of the substance at time t in years.
In the function, e is the mathematical constant known as Euler's number, which is approximately equal to 2.718. The negative sign indicates a decay process, as the quantity of the substance decreases over time.
The decay constant is represented by -1/4. This value determines the rate at which the radioactive substance decays. A higher absolute value of the decay constant means a faster rate of decay.
To find the quantity of the substance at a specific time, substitute the desired value for t in the function Q(t). For example, to find the quantity after 2 years, substitute t = 2:
Q(2) = 100e^(-1/4 * 2)
= 100e^(-1/2)
≈ 100(0.6065)
≈ 60.65
Therefore, the quantity of the radioactive substance after 2 years would be approximately 60.65 units.
To find out how the quantity of a radioactive substance decays over time, we can use the given function:
Q(t) = 100e^(-1/4t)
In this function, Q(t) represents the quantity of the substance at a given time t, and e is the mathematical constant approximately equal to 2.71828, known as Euler's number.
To evaluate this function for a specific time t, follow these steps:
1. Replace the variable t with the given time value.
For example, if we want to find the quantity at t = 5 years, substitute t = 5 into the equation:
Q(5) = 100e^(-1/4 * 5)
2. Simplify the expression inside the parentheses (if needed).
In this case, -1/4 * 5 = -5/4.
Q(5) = 100e^(-5/4)
3. Evaluate the exponential term using a calculator or software that supports the exponential function.
Calculate e^(-5/4), which means raising Euler's number to the power of -5/4:
e^(-5/4) ≈ 0.23254 (rounded to five decimal places)
4. Multiply the result from step 3 by 100 to obtain the quantity at the given time.
Multiply 0.23254 by 100:
Q(5) ≈ 23.254
Therefore, at t = 5 years, the quantity of the radioactive substance is approximately 23.254 units.
To find the step-by-step explanation for the decay of the radioactive substance according to the function Q(t) = 100e^(-1/4t), we need to break down the equation and understand its components.
Step 1: Understand the equation
The equation represents the quantity (Q) of a radioactive substance over time (t) in years. The equation uses the natural exponential function e and a decay constant of -1/4.
Step 2: The decay constant
The decay constant (-1/4) determines the rate at which the radioactive substance decays. Specifically, it indicates that the quantity of the substance will decrease by a factor of e^(-1/4) (approximately 0.778) per year.
Step 3: Initial quantity (Q0)
To find the initial quantity (Q0) of the substance, we would substitute t = 0 into the equation. In this case, Q(0) = 100e^(-1/4 * 0) = 100e^0 = 100. Therefore, the initial quantity of the radioactive substance is 100.
Step 4: Decaying quantity (Q(t))
To find the quantity of the substance at a specific time (t), substitute the given value of t into the equation. For example, if we want to find the quantity at t = 1 year, we would calculate Q(1) = 100e^(-1/4 * 1) = 100e^(-1/4) ≈ 77.8.
Step 5: Interpretation
The equation Q(t) = 100e^(-1/4t) allows us to calculate the quantity of the radioactive substance at any given time (t). As time increases, the quantity will decrease exponentially, approaching zero but never actually reaching zero. The initial quantity of the substance is 100, and as time passes, the quantity will decrease according to the decay constant (-1/4).
Note: It's important to mention that the units of time need to be consistent in this equation. In this case, the unit of time is given in years.