Certainly! Let's calculate the mass of the radioactive element that remains after 150 years, given the half-life of 45 years and an initial mass of 29 grams. The formula to determine the remaining mass [tex]\( y \)[/tex] after [tex]\( t \)[/tex] years is:
[tex]\[
y = 29 \left( \frac{1}{2} \right)^{\frac{t}{45}}
\][/tex]
In this problem, [tex]\( t = 150 \)[/tex] years. We need to substitute the values into the formula and perform the calculations step-by-step.
1. Identify the given values:
- Initial mass, [tex]\( y_0 = 29 \)[/tex] grams.
- Half-life, [tex]\( \text{half-life} = 45 \)[/tex] years.
- Time, [tex]\( t = 150 \)[/tex] years.
2. Substitute the values into the formula:
[tex]\[
y = 29 \left( \frac{1}{2} \right)^{\frac{150}{45}}
\][/tex]
3. Simplify the exponent:
[tex]\[
\frac{150}{45} = \frac{150 \div 15}{45 \div 15} = \frac{10}{3}
\][/tex]
So we rewrite the formula as:
[tex]\[
y = 29 \left( \frac{1}{2} \right)^{\frac{10}{3}}
\][/tex]
4. Calculate the exponentiation:
[tex]\[
\left( \frac{1}{2} \right)^{\frac{10}{3}} \approx 0.099212565748012
\][/tex]
5. Multiply by the initial mass:
[tex]\[
y = 29 \times 0.099212565748012 \approx 2.877164406692361
\][/tex]
So, the remaining mass of the radioactive element after 150 years is approximately [tex]\( 2.877164406692361 \)[/tex] grams.
6. Round the result to three decimal places:
[tex]\[
y \approx 2.877
\][/tex]
Therefore, the remaining mass after 150 years is approximately [tex]\( 2.877 \)[/tex] grams.
