Question 1194826: 1. Suppose there are 350 million people in the US. If the annual birth rate is 2.3% and the annual death rate is 1.9%, what is the relative annual growth rate (r) of the US Population?
a) What will the projected US Population be in 2042 based on an exponential growth model (base e)? Give the model and explain what each part represents.
b) What will the projected US Population be in 2050?
c) What will the projected US Population be in 2100?
d) What would happen if birth and death rates were reversed? Clearly explain what type of model this becomes and how the graph changes.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **1. Calculate the Relative Annual Growth Rate (r)**
* **Net Growth Rate:**
* Birth Rate - Death Rate = 2.3% - 1.9% = 0.4%
* Convert to decimal: 0.4% = 0.004
* **Relative Annual Growth Rate (r):** 0.4% or 0.004
**a) Projected US Population in 2042**
* **Exponential Growth Model:**
* P(t) = P₀ * e^(rt)
* where:
* P(t) is the population at time t
* P₀ is the initial population (350 million)
* r is the annual growth rate (0.004)
* t is the time in years
* e is the base of the natural logarithm (approximately 2.71828)
* **Calculate Population in 2042:**
* t = 2042 - 2024 = 18 years
* P(18) = 350,000,000 * e^(0.004 * 18)
* P(18) ≈ 376,129,370
* **Projected US Population in 2042: Approximately 376 million**
**b) Projected US Population in 2050**
* **Calculate Population in 2050:**
* t = 2050 - 2024 = 26 years
* P(26) = 350,000,000 * e^(0.004 * 26)
* P(26) ≈ 388,360,159
* **Projected US Population in 2050: Approximately 388 million**
**c) Projected US Population in 2100**
* **Calculate Population in 2100:**
* t = 2100 - 2024 = 76 years
* P(76) = 350,000,000 * e^(0.004 * 76)
* P(76) ≈ 474,344,170
* **Projected US Population in 2100: Approximately 474 million**
**d) Reversed Birth and Death Rates**
* **New Growth Rate:** 1.9% (death rate) - 2.3% (birth rate) = -0.4% or -0.004
* **Model Becomes Exponential Decay:** The population would decrease over time.
* **Graph Changes:**
* Instead of an upward curve (exponential growth), the graph would show a downward curve, indicating a declining population.
**Key Points:**
* The exponential growth model assumes a constant growth rate, which may not always be accurate in reality.
* Factors like migration, changes in birth and death rates, and resource availability can influence population growth.
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