Let's solve this question step-by-step using the given data and the calculated results:
### Part (a)
To find the relative change (percent change) in population growth between 2010 and 2011, we use the formula:
[tex]\[
\text{Relative Change} = \left( \frac{\text{Population in 2011} - \text{Population in 2010}}{\text{Population in 2010}} \right) \times 100
\][/tex]
Substitute the given values:
[tex]\[
\text{Relative Change} = \left( \frac{6.998 - 6.914}{6.914} \right) \times 100
\][/tex]
Calculating the relative change:
[tex]\[
\text{Relative Change} \approx 1.2\%
\][/tex]
### Part (b)
Using the percent change calculated in part (a), we can determine the populations for 2012 and 2013.
Population in 2012:
[tex]\[
\text{Population in 2012} = \text{Population in 2011} \times (1 + \frac{\text{Relative Change}}{100})
\][/tex]
[tex]\[
\text{Population in 2012} = 6.998 \times (1 + \frac{1.2}{100}) \approx 7.082 \text{ billion}
\][/tex]
Population in 2013:
[tex]\[
\text{Population in 2013} = \text{Population in 2012} \times (1 + \frac{\text{Relative Change}}{100})
\][/tex]
[tex]\[
\text{Population in 2013} = 7.082 \times (1 + \frac{1.2}{100}) \approx 7.167 \text{ billion}
\][/tex]
### Part (c)
Considering the relative change remains constant each year, we create an equation for the world population [tex]\( P \)[/tex] in billions, [tex]\( x \)[/tex] years after 2010. The initial population in 2010 ([tex]\( P_0 \)[/tex]) is 6.914 billion, and the annual growth rate is given by the relative change:
[tex]\[
P = 6.914 \times (1 + 0.012)^x
\][/tex]
### Part (d)
To predict the world population in 2020 and 2100:
1. Population in 2020: [tex]\( x = 2020 - 2010 = 10 \)[/tex]
2. Population in 2100: [tex]\( x = 2100 - 2010 = 90 \)[/tex]
Using the equation [tex]\( P = 6.914 \times (1 + 0.012)^x \)[/tex]:
For 2020:
[tex]\[
P_{2020} = 6.914 \times (1 + 0.012)^{10} \approx 7.79 \text{ billion}
\][/tex]
For 2100:
[tex]\[
P_{2100} = 6.914 \times (1 + 0.012)^{90} \approx 22.23 \text{ billion}
\][/tex]
Thus, completing the answer boxes in the original question:
- The relative change is [tex]\(1.2\%\)[/tex].
- Population in 2012 is [tex]\(\approx 7.082\)[/tex] billion.
- Population in 2013 is [tex]\(\approx 7.167\)[/tex] billion.
- The equation for predicting population growth is [tex]\( P = 6.914 \cdot (1+0.012)^x \)[/tex].
- The predicted population in 2020 is [tex]\(\approx 7.79\)[/tex] billion.
- The predicted population in 2100 is [tex]\(\approx 22.23\)[/tex] billion.
