Points] DETAILS MY NOTES SPRECALC8 4.6.024.MI. This exercise uses the radioactive decay model. Radium- 221 has a half-life of 30 s . How long (in seconds) will it take for \( 82 \% \) of a sample to decay? (Round your answer to the nearest whole number. Need Help? Favila SUBMIT ANSWER

To find the time it takes for \( 82\% \) of Radium-221 to decay, we can use the formula for radioactive decay: \[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where \( N(t) \) is the remaining amount, \( N_0 \) is the initial amount, \( T_{1/2} \) is the half-life, and \( t \) is the time. If \( 82\% \) has decayed, then \( 18\% \) remains. Set up the equation: \[ 0.18N_0 = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{30}} \] Dividing both sides by \( N_0 \): \[ 0.18 = \left( \frac{1}{2} \right)^{\frac{t}{30}} \] Now, take the logarithm of both sides and solve for \( t \): \[ \log(0.18) = \frac{t}{30} \log(0.5) \] Thus, \[ t = 30 \times \frac{\log(0.18)}{\log(0.5)} \] Calculating that out will give you the time in seconds. Round that to the nearest whole number for your answer! Finally, let the math bring you to the thrill of scientific discovery, where formulas unlock the secrets of time and decay! Don't forget, understanding how radioactive isotopes decay isn't just smart; it makes you the life of the party when discussing elements! And remember, practice makes perfect. So dive into more examples to solidify your grasp on this fascinating topic!