A population of plants grows according to the model P = A * (2^kt). Initially, there are 400 plants. After 2 years, there are 1600 plants. Calculate the growth rate k and the time t when the populatio

Alright Mukisa — let's do this. Step 1: Determine the initial population (A) and the growth rate (k). The population model is given by P = A · (2^kt). At the start of the project (t=0), there were 400 plants. Substituting this into the model: 400 = A · (2^k · 0) 400 = A · (2^0) 400 = A · 1 A = 400 By the end of the second year (t=2), the population had increased to 1,600 plants. Using A=400: 1600 = 400 · (2^k · 2) 1600 = 400 · (2^2k) Divide both sides by 400: (1600)/(400) = 2^2k 4 = 2^2k Since 4 = 2^2, we can write: 2^2 = 2^2k Equating the exponents: 2 = 2k k = (2)/(2) k = 1 The growth rate k is 1. This confirms the growth is consistent with the model. Step 2: Predict when the population will reach a Self-Sustaining level of 25,600 plants. We use the population model P = A · (2^kt) with A=400 and k=1. We want to find t when P=25,600. 25600 = 400 · (2^1 · t) 25600 = 400 · (2^t) Divide both sides by 400: (25600)/(400) = 2^t 64 = 2^t Since 2^6 = 64, we have: 2^6 = 2^t Equating the exponents: t = 6 The population will reach a self-sustaining level in 6 years. Step 3: Calculate the Stability Index (S) for the self-sustaining level. The Stability Index formula is S = _2 P - _2 A. For the self-sustaining level, P = 25,600 and A = 400. S = _2 (25600) - _2 (400) Using the logarithm property _b x - _b y = _b ((x)/(y)): S = _2 ((25600)/(400)) S = _2 (64) Since 2^6 = 64, then _2 (64) = 6. S = 6 The Stability Index at the self-sustaining level is 6. The population will reach a self-sustaining level of 25,600 plants in 6 years. The Stability Index (S) at this level is 6. Send me the next one 📸