Final answer:
a) The population in 1925 was approximately 40.801 million. b) The population will be 12,000,000 in approximately 27.474 years from now. c) The maximum sustainable population is approximately 971.362 million. d) Not enough information to determine.
Step-by-step explanation:
a) To find the population in 1925, substitute t = 1925 - 1800 = 125 into the equation P(t) = 17.25 * (1 + 55.325 * e^(-0.0295t)).
P(125) = 17.25 * (1 + 55.325 * e^(-0.0295 * 125)) = 17.25 * (1 + 55.325 * e^(-3.6875)) ≈ 17.25 * (1 + 55.325 * 0.0247) ≈ 17.25 * (1 + 1.362) = 17.25 * 2.362 ≈ 40.801 million.
Therefore, the population in 1925 was approximately 40.801 million.
b) To find when the population will be 12,000,000, substitute P(t) = 12 into the equation and solve for t.
12 = 17.25 * (1 + 55.325 * e^(-0.0295t))
(1 + 55.325 * e^(-0.0295t)) = 12/17.25
55.325 * e^(-0.0295t) = 12/17.25 - 1
e^(-0.0295t) = (12/17.25 - 1)/55.325
Take the natural logarithm of both sides: -0.0295t = ln((12/17.25 - 1)/55.325)
t = ln((12/17.25 - 1)/55.325) / -0.0295 ≈ 27.474 years.
Therefore, the population will be 12,000,000 in approximately 27.474 years from now.
c) The maximum sustainable population is the population at which P(t) reaches its maximum value. To find this, we need to find the maximum value of the function. Since the function is continuously decreasing, the maximum value occurs at the lower limit of t, which is t = 0.
P(0) = 17.25 * (1 + 55.325 * e^(-0.0295 * 0)) = 17.25 * (1 + 55.325 * e^0) = 17.25 * (1 + 55.325 * 1) = 17.25 * 56.325 ≈ 971.362 million.
Therefore, the maximum sustainable population is approximately 971.362 million.
d) Not enough information to determine.