This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Identify the given information and the goal. The half-life (t_1/2) of the radioactive isotope is 15 days. We need to find the time (t) it takes for 90% of the isotope to decay. This means 10% of the isotope remains. Step 2: Relate the remaining amount to the initial amount. If 90% of the isotope has decayed, then the fraction remaining is 100\% - 90\% = 10\%, or 0.10. So, (N_t)/(N_0) = 0.10, where N_t is the amount remaining at time t, and N_0 is the initial amount. Step 3: Use the half-life formula. The formula relating the amount remaining to the initial amount and time is: N_t = N_0 ((1)/(2))^t/t_1/2 Divide by N_0: (N_t)/(N_0) = ((1)/(2))^t/t_1/2 Step 4: Substitute the known values into the formula. 0.10 = ((1)/(2))^t/15 Step 5: Solve for t by taking the logarithm of both sides. Take the natural logarithm (ln) of both sides: (0.10) = (((1)/(2))^t/15) Using the logarithm property (a^b) = b (a): (0.10) = (t)/(15) ((1)/(2)) We know that ((1)/(2)) = (1) - (2) = 0 - (2) = -(2). So, (0.10) = (t)/(15) (-(2)) Step 6: Isolate t and calculate its value. t = 15 × ((0.10))/(-(2)) t = 15 × (-2.302585)/(-0.693147) t = 15 × 3.3219 t = 49.8285 Rounding to a reasonable number of significant figures: t ≈ 49.8 days The time it will take for 90% of the isotope to decay is approximately 49.8 days. Drop the next question.