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
Okay, MEDIC, let's break down these half-life problems step by step. Question 1: 400g decayed to 50g in 24 days. Find the half-life t_(1)/(2). The formula for radioactive decay is: N(t) = N_0 ((1)/(2))^(t)/(T) where N(t) is the final mass, N_0 is the initial mass, t is the total time, and T is the half-life. Step 1: Identify the given values. N_0 = 400 g N(t) = 50 g t = 24 days We need to find T. Step 2: Substitute the values into the decay formula. 50 = 400 ((1)/(2))^(24)/(T) Step 3: Isolate the term with the exponent. Divide both sides by 400. (50)/(400) = ((1)/(2))^(24)/(T) (1)/(8) = ((1)/(2))^(24)/(T) Step 4: Express (1)/(8) as a power of (1)/(2). Since 8 = 2^3, then (1)/(8) = ((1)/(2))^3. ((1)/(2))^3 = ((1)/(2))^(24)/(T) Step 5: Equate the exponents since the bases are the same. 3 = (24)/(T) Step 6: Solve for T. T = (24)/(3) T = 8 days The half-life is 8 days. --- Question 2: A substance has half-life = 4 years. Original mass = 256g. After how many years will only 16g be left? Find t. The formula for radioactive decay is: N(t) = N_0 ((1)/(2))^(t)/(T) Step 1: Identify the given values. T = 4 years N_0 = 256 g N(t) = 16 g We need to find t. Step 2: Substitute the values into the decay formula. 16 = 256 ((1)/(2))^(t)/(4) Step 3: Isolate the term with the exponent. Divide both sides by 256. (16)/(256) = ((1)/(2))^(t)/(4) (1)/(16) = ((1)/(2))^(t)/(4) Step 4: Express (1)/(16) as a power of (1)/(2). Since 16 = 2^4, then (1)/(16) = ((1)/(2))^4. ((1)/(2))^4 = ((1)/(2))^(t)/(4) Step 5: Equate the exponents since the bases are the same. 4 = (t)/(4) Step 6: Solve for t. t = 4 × 4 t = 16 years The time elapsed is 16 years. --- Question 3: Half-life = 3 hours. 512g left after 12 hours. Find the Original mass N_0. The formula for radioactive decay is: N(t) = N_0 ((1)/(2))^(t)/(T) Step 1: Identify the given values. T = 3 hours N(t) = 512 g t = 12 hours We need to find N_0. Step 2: Substitute the values into the decay formula. 512 = N_0 ((1)/(2))^(12)/(3) Step 3: Simplify the exponent. 512 = N_0 ((1)/(2))^4 Step 4: Calculate the value of ((1)/(2))^4. ((1)/(2))^4 = (1^4)/(2^4) = (1)/(16) Step 5: Substitute this value back into the equation. 512 = N_0 × (1)/(16) Step 6: Solve for N_0. Multiply both sides by 16. N_0 = 512 × 16 N_0 = 8192 g The original mass was 8192 g. 3 done, 2 left today. You're making progress.