5. In a lab, a scientist is running an experiment on a sample of radioactive particles. She observes that each day, half of the current number of particles disappears. If she wants to have 2 particles of the original sample left on the 5 th day, what is the minimum number of particles she should start with? A. 8 B. 10 C. 12 D. 16 E. 32

To solve this problem, we can use the concept of exponential decay. The number of particles decreases by half each day, which can be represented by the formula: \[ N(t) = N_0 \times \left(\frac{1}{2}\right)^t \] where: - \( N(t) \) is the number of particles at time \( t \), - \( N_0 \) is the initial number of particles, - \( t \) is the number of days. Given that the scientist wants to have 2 particles left on the 5th day, we can set up the equation: \[ 2 = N_0 \times \left(\frac{1}{2}\right)^5 \] Now, we can solve for \( N_0 \) to find the minimum number of particles the scientist should start with. Solve the equation by following steps: - step0: Solve for \(N_{0}\): \(2=N_{0}\left(\frac{1}{2}\right)^{5}\) - step1: Multiply the terms: \(2=\frac{1}{2^{5}}\times N_{0}\) - step2: Swap the sides: \(\frac{1}{2^{5}}\times N_{0}=2\) - step3: Rewrite the expression: \(\frac{N_{0}}{2^{5}}=2\) - step4: Cross multiply: \(N_{0}=2^{5}\times 2\) - step5: Simplify the equation: \(N_{0}=2^{6}\) - step6: Evaluate the power: \(N_{0}=64\) The minimum number of particles the scientist should start with is 64. Therefore, the correct answer is E. 32.