__ A sample of radium has a weight of 1.5 mg and a half-life of approximately 6 years. 1) How much of the sample will remain after: iii) 1 year? i) 6 years? ii) 3 years?

Here's how to calculate the remaining radium after a given time, using the concept of half-life:<br /><br />**Understanding Half-life**<br /><br />A half-life is the time it takes for half of a radioactive substance to decay. After one half-life, 50% remains. After two half-lives, 25% remains (half of the remaining 50%), and so on.<br /><br />**Formula**<br /><br />The formula for radioactive decay is:<br /><br />*N(t) = N₀ * (1/2)^(t/T)*<br /><br />Where:<br /><br />* N(t) = the amount remaining after time t<br />* N₀ = the initial amount<br />* t = the time that has passed<br />* T = the half-life<br /><br />**Calculations**<br /><br />In this case, N₀ = 1.5 mg and T = 6 years.<br /><br />**i) 6 years (1 half-life)**<br /><br />* N(6) = 1.5 * (1/2)^(6/6)<br />* N(6) = 1.5 * (1/2)^1<br />* N(6) = 1.5 * 0.5<br />* N(6) = 0.75 mg<br /><br />**ii) 3 years (0.5 half-lives)**<br /><br />* N(3) = 1.5 * (1/2)^(3/6)<br />* N(3) = 1.5 * (1/2)^0.5<br />* N(3) = 1.5 * 0.7071 (approximately)<br />* N(3) ≈ 1.06 mg<br /><br />**iii) 1 year (1/6 of a half-life)**<br /><br />* N(1) = 1.5 * (1/2)^(1/6)<br />* N(1) = 1.5 * 0.8909 (approximately)<br />* N(1) ≈ 1.34 mg<br /><br /><br />**Summary of Results**<br /><br />* After 1 year: Approximately 1.34 mg will remain.<br />* After 3 years: Approximately 1.06 mg will remain.<br />* After 6 years: 0.75 mg will remain.<br />