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Math  /  Algebra

QuestionSuppose the amount of a certain radioactive substance in a sample decays from 5.20 mg to 1.90 mg over a period of 6.21 minutes. Calculate the half life of the substance.
Round your answer to 2 significant digits. □\square □\square ×10\times 10

Studdy Solution

STEP 1

1. The decay of the radioactive substance follows an exponential decay model.
2. The initial amount of the substance is 5.20 mg.
3. The final amount of the substance after 6.21 minutes is 1.90 mg.
4. We need to calculate the half-life of the substance.

STEP 2

1. Use the exponential decay formula.
2. Solve for the decay constant.
3. Calculate the half-life using the decay constant.
4. Round the answer to 2 significant digits.

STEP 3

Use the exponential decay formula:
The formula for exponential decay is given by: A(t)=A0⋅e−kt A(t) = A_0 \cdot e^{-kt}
Where: - A(t) A(t) is the amount of substance at time t t . - A0 A_0 is the initial amount of substance. - k k is the decay constant. - t t is the time elapsed.
Given: - A0=5.20 A_0 = 5.20 mg - A(t)=1.90 A(t) = 1.90 mg - t=6.21 t = 6.21 minutes

STEP 4

Solve for the decay constant k k .
Rearrange the formula to solve for k k : 1.90=5.20⋅e−6.21k 1.90 = 5.20 \cdot e^{-6.21k}
Divide both sides by 5.20: 1.905.20=e−6.21k \frac{1.90}{5.20} = e^{-6.21k}
Take the natural logarithm of both sides: ln⁡(1.905.20)=−6.21k \ln\left(\frac{1.90}{5.20}\right) = -6.21k
Solve for k k : k=−ln⁡(1.905.20)6.21 k = -\frac{\ln\left(\frac{1.90}{5.20}\right)}{6.21}

STEP 5

Calculate the half-life using the decay constant.
The formula for half-life T1/2 T_{1/2} is: T1/2=ln⁡(2)k T_{1/2} = \frac{\ln(2)}{k}
Substitute the value of k k obtained in STEP_2 into the formula.

STEP 6

Round the answer to 2 significant digits.
Calculate T1/2 T_{1/2} and round to 2 significant digits.
T1/2≈3.2 minutes T_{1/2} \approx 3.2 \text{ minutes}

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