Questions: The half-life of radium-226 is 1600 years. Suppose we have a 26-mg sample. (a) Find the yearly growth factor a. (Round your answer to eight decimal places.) a= (b) Find an exponential model m(t)=C a^t for the mass remaining after t years. m(t)= (c) How much of the sample will remain after 3900 years? (Round your answer to two decimal places.) mg (d) After how long will only 18 mg of the sample remain? (Round your answer to the nearest year.) yr

Solution Steps

(a) To find the yearly growth factor \( a \), use the formula for half-life: \( a^{1600} = \frac{1}{2} \). Solve for \( a \) by taking the 1600th root of \( \frac{1}{2} \).

(b) The exponential model for the mass remaining after \( t \) years is given by \( m(t) = C \cdot a^t \), where \( C \) is the initial mass and \( a \) is the yearly growth factor found in part (a).

(c) To find how much of the sample will remain after 3900 years, substitute \( t = 3900 \) into the exponential model \( m(t) = C \cdot a^t \) and calculate the result.

To find the yearly growth factor \( a \), we use the half-life formula: \[ a^{1600} = \frac{1}{2} \] Taking the 1600th root of both sides, we find: \[ a = \left(\frac{1}{2}\right)^{\frac{1}{1600}} \approx 0.99956688 \]

The exponential model for the mass remaining after \( t \) years is given by: \[ m(t) = C \cdot a^t \] where \( C = 26 \) mg is the initial mass. Thus, the model becomes: \[ m(t) = 26 \cdot (0.99956688)^t \]

To find the mass remaining after 3900 years, we substitute \( t = 3900 \) into the model: \[ m(3900) = 26 \cdot (0.99956688)^{3900} \approx 4.8 \text{ mg} \]

Final Answer