This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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a) Step 1: Set up the exponential growth model. The exponential growth model is given by P(t) = P_0 e^kt, where P(t) is the population at time t, P_0 is the initial population, and k is the growth rate. Given: P_0 = 10000 (population in 1960, which we take as t=0) P(10) = 12000 (population in 1970, so t=1970-1960=10 years) Step 2: Calculate the growth rate k. Substitute the values into the model: 12000 = 10000 e^k · 10 Divide by 10000: (12000)/(10000) = e^10k 1.2 = e^10k Take the natural logarithm of both sides: (1.2) = (e^10k) (1.2) = 10k k = ((1.2))/(10) Step 3: Estimate the population in 1980. The time from 1960 to 1980 is t = 1980 - 1960 = 20 years. Substitute k and t=20 into the model: P(20) = 10000 e^(((1.2))/(10)) · 20 P(20) = 10000 e^2 (1.2) Using the logarithm property a b = b^a: P(20) = 10000 e^((1.2)^2) Using the property e^ x = x: P(20) = 10000 × (1.2)^2 P(20) = 10000 × 1.44 P(20) = 14400 The estimated population in 1980 is 14400. b) Step 1: Define doubling time. The doubling time T_d is the time it takes for the population to double. If the initial population is P_0, the population after T_d years will be 2P_0. Using the exponential growth model P(t) = P_0 e^kt: 2P_0 = P_0 e^kT_d 2 = e^kT_d Step 2: Solve for T_d. Take the natural logarithm of both sides: 2 = (e^kT_d) 2 = kT_d T_d = ( 2)/(k) From part (a), we found k = ((1.2))/(10). Substitute this value: T_d = ( 2)/((1.2))10 T_d = (10 2)/((1.2)) Step 3: Calculate the numerical value for T_d. Using a calculator: 2 ≈ 0.6931 1.2 ≈ 0.1823 T_d ≈ (10 × 0.6931)/(0.1823) T_d ≈ (6.931)/(0.1823) T_d ≈ 38.01 years The doubling time for the town's population is approximately 38.01 years. c) Step 1: Set up the identity. We want to express x + sqrt(3) x in the form R (x+). Using the angle addition formula for sine, R (x+) = R( x + x ). R (x+) = (R ) x + (R ) x Comparing this to x + sqrt(3) x: R = 1 (Equation 1) R = sqrt(3) (Equation 2) Step 2: Find R. Square both equations and add them: (R )^2 + (R )^2 = 1^2 + (sqrt(3))^2 R^2 ^2 + R^2 ^2 = 1 + 3 R^2 (^2 + ^2 ) = 4 Since ^2 + ^2 = 1: R^2 (1) = 4 R = sqrt(4) Since R represents an amplitude, it must be positive: R = 2 Step 3: Find . Divide Equation 2 by Equation 1: (R )/(R ) = sqrt(3)1 = sqrt(3) Since R = 1 > 0 and R = sqrt(3) > 0, is in the first quadrant. = (sqrt(3)) = ()/(3) radians or 60^ So, x + sqrt(3) x = 2 (x + ()/(3)). The values are R=2 and =()/(3) radians or 60^. 3 done, 2 left today. You're making progress.