Homework \# I Score: $9.59 / 18 \quad 12 / 18$ answered Question 17 The half-life of Radium-226 is 1590 yetrs. If a sample contains $400 \mathrm{mg}$, how many mg will remain after 4000 years? $\mathrm{mg}$ Give your answer accurate to at least 2 decimal places. Question Help: $\square$ Video $1 \square$ Video 2 Submit Question

Step 1 ：The problem is asking for the remaining amount of Radium-226 after 4000 years given its half-life is 1590 years and the initial amount is 400 mg.

Step 2 ：The formula to calculate the remaining amount of a substance after a certain time given its half-life is: \(N = N0 * (1/2)^{t/T}\) where: \(N\) is the final amount remaining after the time given, \(N0\) is the initial amount, \(t\) is the time that has passed, \(T\) is the half-life of the substance.

Step 3 ：In this case, \(N0 = 400\) mg, \(t = 4000\) years, and \(T = 1590\) years.

Step 4 ：We can substitute these values into the formula to find \(N\), the remaining amount of Radium-226 after 4000 years.

Step 5 ：\(N0 = 400\)

Step 6 ：\(t = 4000\)

Step 7 ：\(T = 1590\)

Step 8 ：\(N = 69.94421915270917\)

Step 9 ：The remaining amount of Radium-226 after 4000 years is approximately \(\boxed{69.94 \, \text{mg}}\).