Chapter 3 - Section 3.8 - Exponential Growth and Decay - 3.8 Exercises - Page 243: 5

(a) $P(150) = 1508$ In 1900, the actual population (in millions) was 1650, so the model's prediction is a bit less than the actual population. $P(200) = 1871$ In 1950, the actual population (in millions) was 2560, so the model's prediction is less than the actual population. (b) $P(100) = 2161$ In 1950, the actual population (in millions) was 2560, so the model's prediction is less than the actual population. (c) $P(100) = 3972$ In 2000, the actual population (in millions) was 6080, so the model's prediction is significantly less than the actual population.

(a) We can find the value of $k$: $P(t) = P(0)e^{kt}$ $P(50) = 790~e^{50k} = 980$ $e^{50k} = \frac{980}{790}$ $50k = ln(\frac{980}{790})$ $k = \frac{ln(\frac{980}{790})}{50}$ $k = 0.00431$ We can find $P(150)$: $P(t) = P(0)e^{kt}$ $P(150) = 790~e^{(0.00431)(150)}$ $P(150) = 1508$ In 1900, the actual population (in millions) was 1650, so the model's prediction is a bit less than the actual population. We can find $P(200)$: $P(t) = P(0)e^{kt}$ $P(200) = 790~e^{(0.00431)(200)}$ $P(200) = 1871$ In 1950, the actual population (in millions) was 2560, so the model's prediction is less than the actual population. (b) We can find the value of $k$: $P(t) = P(0)e^{kt}$ $P(50) = 1260~e^{50k} = 1650$ $e^{50k} = \frac{1650}{1260}$ $50k = ln(\frac{1650}{1260})$ $k = \frac{ln(\frac{1650}{1260})}{50}$ $k = 0.00539327$ We can find $P(100)$: $P(t) = P(0)e^{kt}$ $P(100) = 1260~e^{(0.00539327)(100)}$ $P(100) = 2161$ In 1950, the actual population (in millions) was 2560, so the model's prediction is less than the actual population. (c) We can find the value of $k$: $P(t) = P(0)e^{kt}$ $P(50) = 1650~e^{50k} = 2560$ $e^{50k} = \frac{2560}{1650}$ $50k = ln(\frac{2560}{1650})$ $k = \frac{ln(\frac{2560}{1650})}{50}$ $k = 0.0087846$ We can find $P(100)$: $P(t) = P(0)e^{kt}$ $P(100) = 1650~e^{(0.0087846)(100)}$ $P(100) = 3972$ In 2000, the actual population (in millions) was 6080, so the model's prediction is significantly less than the actual population. The period between 1950 and 2000 saw an increase in the relative rate of global population growth. This could be explained by numerous factors including better health and medical care which led to both a higher percentage of children surviving the first few years of life and a longer life expectancy,