Step 1 of 3 The halflife of titanium51 is approximately 576

Step 1 of 3: The half-life of titanium-51 is approximately 5.76 minutes. Determine a so that A(t) = A₀ a ^t describes the amount of titanium-51 left after t minutes, where A ₀ is the amount at time t = 0 . Round to six decimal places.

Solution

The half life of Titanium-51 is 5.76 minutes approximately. Further, A = A₀a^t where A₀ is the initial amount of Titanium-51 when t = 0 and A(t) is the amount of Titanium-51 left after t minutes. Then A₀/2 = A₀ a^5.76   or, on dividing both the sides by A₀, we have ½ = a^5.76 . Now, on taking logarithm of both the sides, we have log (1/2) = 5.76 log a (as log x^y = y logx) or, -0.301029995 = 5.76 log a or, log a = -0.301029995 /5.76 = -0.05226152. Hence a = 10^-0.05226152 = 0.886621952 = 0.886622 (on rounding off to 6 decimal places).