Answer :
Certainly! Let's go through each part of the question step-by-step to find the average rate of change for each population over the specified intervals.
Given:
Table 1.12
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \text{Year} & 1990 & 1992 & 1995 & 2000 & 2007 \\ \hline P_1 & 53 & 63 & 73 & 83 & 93 \\ \hline P_2 & 85 & 80 & 75 & 70 & 65 \\ \hline \end{array} \][/tex]
### (a) Average Rate of Change from 1990 to 2000
For [tex]\( P_1 \)[/tex]:
1. Initial population ([tex]\( P_1 \)[/tex]) in 1990: 53
2. Population ([tex]\( P_1 \)[/tex]) in 2000: 83
3. Change in population: [tex]\( 83 - 53 = 30 \)[/tex]
4. Change in years: [tex]\( 2000 - 1990 = 10 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_1 \)[/tex] is:
[tex]\[ \frac{30}{10} = 3.0 \][/tex]
For [tex]\( P_2 \)[/tex]:
1. Initial population ([tex]\( P_2 \)[/tex]) in 1990: 85
2. Population ([tex]\( P_2 \)[/tex]) in 2000: 70
3. Change in population: [tex]\( 70 - 85 = -15 \)[/tex]
4. Change in years: [tex]\( 2000 - 1990 = 10 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_2 \)[/tex] is:
[tex]\[ \frac{-15}{10} = -1.5 \][/tex]
### (b) Average Rate of Change from 1995 to 2007
For [tex]\( P_1 \)[/tex]:
1. Initial population ([tex]\( P_1 \)[/tex]) in 1995: 73
2. Population ([tex]\( P_1 \)[/tex]) in 2007: 93
3. Change in population: [tex]\( 93 - 73 = 20 \)[/tex]
4. Change in years: [tex]\( 2007 - 1995 = 12 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_1 \)[/tex] is:
[tex]\[ \frac{20}{12} \approx 1.67 \][/tex]
For [tex]\( P_2 \)[/tex]:
1. Initial population ([tex]\( P_2 \)[/tex]) in 1995: 75
2. Population ([tex]\( P_2 \)[/tex]) in 2007: 65
3. Change in population: [tex]\( 65 - 75 = -10 \)[/tex]
4. Change in years: [tex]\( 2007 - 1995 = 12 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_2 \)[/tex] is:
[tex]\[ \frac{-10}{12} \approx -0.83 \][/tex]
### (c) Average Rate of Change from 1990 to 2007
For [tex]\( P_1 \)[/tex]:
1. Initial population ([tex]\( P_1 \)[/tex]) in 1990: 53
2. Population ([tex]\( P_1 \)[/tex]) in 2007: 93
3. Change in population: [tex]\( 93 - 53 = 40 \)[/tex]
4. Change in years: [tex]\( 2007 - 1990 = 17 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_1 \)[/tex] is:
[tex]\[ \frac{40}{17} \approx 2.35 \][/tex]
For [tex]\( P_2 \)[/tex]:
1. Initial population ([tex]\( P_2 \)[/tex]) in 1990: 85
2. Population ([tex]\( P_2 \)[/tex]) in 2007: 65
3. Change in population: [tex]\( 65 - 85 = -20 \)[/tex]
4. Change in years: [tex]\( 2007 - 1990 = 17 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_2 \)[/tex] is:
[tex]\[ \frac{-20}{17} \approx -1.18 \][/tex]
### Summary
The average rates of change for the two populations over the specified intervals are:
- From 1990 to 2000:
- [tex]\( P_1 \)[/tex]: [tex]\( 3.0 \)[/tex]
- [tex]\( P_2 \)[/tex]: [tex]\( -1.5 \)[/tex]
- From 1995 to 2007:
- [tex]\( P_1 \)[/tex]: [tex]\( 1.67 \)[/tex]
- [tex]\( P_2 \)[/tex]: [tex]\( -0.83 \)[/tex]
- From 1990 to 2007:
- [tex]\( P_1 \)[/tex]: [tex]\( 2.35 \)[/tex]
- [tex]\( P_2 \)[/tex]: [tex]\( -1.18 \)[/tex]
Given:
Table 1.12
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \text{Year} & 1990 & 1992 & 1995 & 2000 & 2007 \\ \hline P_1 & 53 & 63 & 73 & 83 & 93 \\ \hline P_2 & 85 & 80 & 75 & 70 & 65 \\ \hline \end{array} \][/tex]
### (a) Average Rate of Change from 1990 to 2000
For [tex]\( P_1 \)[/tex]:
1. Initial population ([tex]\( P_1 \)[/tex]) in 1990: 53
2. Population ([tex]\( P_1 \)[/tex]) in 2000: 83
3. Change in population: [tex]\( 83 - 53 = 30 \)[/tex]
4. Change in years: [tex]\( 2000 - 1990 = 10 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_1 \)[/tex] is:
[tex]\[ \frac{30}{10} = 3.0 \][/tex]
For [tex]\( P_2 \)[/tex]:
1. Initial population ([tex]\( P_2 \)[/tex]) in 1990: 85
2. Population ([tex]\( P_2 \)[/tex]) in 2000: 70
3. Change in population: [tex]\( 70 - 85 = -15 \)[/tex]
4. Change in years: [tex]\( 2000 - 1990 = 10 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_2 \)[/tex] is:
[tex]\[ \frac{-15}{10} = -1.5 \][/tex]
### (b) Average Rate of Change from 1995 to 2007
For [tex]\( P_1 \)[/tex]:
1. Initial population ([tex]\( P_1 \)[/tex]) in 1995: 73
2. Population ([tex]\( P_1 \)[/tex]) in 2007: 93
3. Change in population: [tex]\( 93 - 73 = 20 \)[/tex]
4. Change in years: [tex]\( 2007 - 1995 = 12 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_1 \)[/tex] is:
[tex]\[ \frac{20}{12} \approx 1.67 \][/tex]
For [tex]\( P_2 \)[/tex]:
1. Initial population ([tex]\( P_2 \)[/tex]) in 1995: 75
2. Population ([tex]\( P_2 \)[/tex]) in 2007: 65
3. Change in population: [tex]\( 65 - 75 = -10 \)[/tex]
4. Change in years: [tex]\( 2007 - 1995 = 12 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_2 \)[/tex] is:
[tex]\[ \frac{-10}{12} \approx -0.83 \][/tex]
### (c) Average Rate of Change from 1990 to 2007
For [tex]\( P_1 \)[/tex]:
1. Initial population ([tex]\( P_1 \)[/tex]) in 1990: 53
2. Population ([tex]\( P_1 \)[/tex]) in 2007: 93
3. Change in population: [tex]\( 93 - 53 = 40 \)[/tex]
4. Change in years: [tex]\( 2007 - 1990 = 17 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_1 \)[/tex] is:
[tex]\[ \frac{40}{17} \approx 2.35 \][/tex]
For [tex]\( P_2 \)[/tex]:
1. Initial population ([tex]\( P_2 \)[/tex]) in 1990: 85
2. Population ([tex]\( P_2 \)[/tex]) in 2007: 65
3. Change in population: [tex]\( 65 - 85 = -20 \)[/tex]
4. Change in years: [tex]\( 2007 - 1990 = 17 \)[/tex]
The average rate of change [tex]\( \frac{\text{Change in population}}{\text{Change in years}} \)[/tex] for [tex]\( P_2 \)[/tex] is:
[tex]\[ \frac{-20}{17} \approx -1.18 \][/tex]
### Summary
The average rates of change for the two populations over the specified intervals are:
- From 1990 to 2000:
- [tex]\( P_1 \)[/tex]: [tex]\( 3.0 \)[/tex]
- [tex]\( P_2 \)[/tex]: [tex]\( -1.5 \)[/tex]
- From 1995 to 2007:
- [tex]\( P_1 \)[/tex]: [tex]\( 1.67 \)[/tex]
- [tex]\( P_2 \)[/tex]: [tex]\( -0.83 \)[/tex]
- From 1990 to 2007:
- [tex]\( P_1 \)[/tex]: [tex]\( 2.35 \)[/tex]
- [tex]\( P_2 \)[/tex]: [tex]\( -1.18 \)[/tex]
