The number of bacteria in a culture is increasing according to the law of exponential growth. After 2 hours, there are 50 bacteria, and after 4 hours, there are 200 bacteria. How many bacteria will there after 6 hours? 400 Need Help? Racteria SUBMIT ANSWEA

Let the number of bacteria at time t be given by the exponential growth model   N(t) = N₀ · e^(kt), where N₀ is the initial number (at time t = 0) and k is the constant growth rate. We are given:  • N(2) = 50  • N(4) = 200 Step 1. Write the equations for t = 2 and t = 4:   N(2) = N₀ · e^(2k) = 50   (1)   N(4) = N₀ · e^(4k) = 200   (2) Step 2. Divide equation (2) by equation (1) to eliminate N₀:   [N₀ · e^(4k)] / [N₀ · e^(2k)] = 200/50 Simplify:   e^(4k − 2k) = 4   e^(2k) = 4 Step 3. Solve for k:   Take the natural logarithm of both sides:   2k = ln 4   k = (ln 4) / 2 Step 4. We want the number after 6 hours, N(6):   N(6) = N₀ · e^(6k) To find N₀, use equation (1):   N₀ = 50 / e^(2k) Now substitute back into the equation for N(6):   N(6) = [50 / e^(2k)] · e^(6k) = 50 · e^(4k) Since we already found that e^(2k) = 4, then:   e^(4k) = (e^(2k))² = 4² = 16 Thus:   N(6) = 50 · 16 = 800 Therefore, after 6 hours, there will be 800 bacteria.