The mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of per day. Find the half-life of this
substance (that is, the time it takes for one-half the original amount in a given sample of this substance to decay).
Note: This is a continuous exponential decay model.
Do not round any intermediate computations, and round your answer to the nearest hundredth.
days

Question

The mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of

per day. Find the half-life of this
substance (that is, the time it takes for one-half the original amount in a given sample of this substance to decay).
Note: This is a continuous exponential decay model.
Do not round any intermediate computations, and round your answer to the nearest hundredth.
days

UpStudy AI · Accepted Answer

To find the half-life of a radioactive substance that follows a continuous exponential decay model, we can use the formula for exponential decay:

$$ A(t) = A_0 \times e^{-kt} $$

where:
- $$ A(t) $$ is the amount of the substance at time $$ t $$,
- $$ A_0 $$ is the initial amount of the substance,
- $$ k $$ is the decay rate parameter,
- $$ t $$ is the time.

Given that the decay rate parameter $$ k $$ is $$ 5.8\% $$ per day, we can convert this to a decimal by dividing by 100:

$$ k = 5.8\% = 0.058 $$

We want to find the half-life, which is the time it takes for one-half the original amount to decay. This means that at the half-life, $$ A(t) = \frac{1}{2}A_0 $$.

Substitute $$ A(t) = \frac{1}{2}A_0 $$ into the exponential decay formula:

$$ \frac{1}{2}A_0 = A_0 \times e^{-0.058t} $$

Simplify the equation:

$$ \frac{1}{2} = e^{-0.058t} $$

To solve for $$ t $$, we can take the natural logarithm of both sides:

$$ \ln\left(\frac{1}{2}\right) = \ln(e^{-0.058t}) $$

$$ \ln\left(\frac{1}{2}\right) = -0.058t $$

Now, we can solve for $$ t $$:

$$ t = \frac{\ln\left(\frac{1}{2}\right)}{-0.058} $$

Let's calculate the half-life using this formula.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\ln{\left(\frac{1}{2}\right)}}{-0.058}$$
- step1: Simplify:
  $$\frac{-\ln{\left(2\right)}}{-0.058}$$
- step2: Rewrite the expression:
  $$\frac{-\ln{\left(2\right)}}{-\frac{29}{500}}$$
- step3: Multiply by the reciprocal:
  $$-\ln{\left(2\right)}\times \left(-\frac{500}{29}\right)$$
- step4: Multiply the terms:
  $$\ln{\left(2\right)}\times \frac{500}{29}$$
- step5: Multiply the terms:
  $$\frac{\ln{\left(2\right)}\times 500}{29}$$
- step6: Multiply the terms:
  $$\frac{500\ln{\left(2\right)}}{29}$$
The half-life of the radioactive substance is approximately 11.95 days.
The half-life of the radioactive substance is approximately 11.95 days.

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