Let's thoroughly evaluate the percentage growth rates for each base value provided, step-by-step.
### Basic Understanding:
The weekly percent growth rate can be determined using the formula:
[tex]\[
\text{Growth Rate (\%)} = (b - 1) \times 100
\][/tex]
where [tex]\( b \)[/tex] is the given base value.
### Calculate Growth Rates for Each Base Value:
1. For [tex]\( b = 1.8 \)[/tex]:
[tex]\[
\text{Growth Rate} = (1.8 - 1) \times 100 = 0.8 \times 100 = 80\%
\][/tex]
The weekly percent growth rate is [tex]\( 80\% \)[/tex].
2. For [tex]\( b = 106 \)[/tex]:
[tex]\[
\text{Growth Rate} = (106 - 1) \times 100 = 105 \times 100 = 10500\%
\][/tex]
The weekly percent growth rate is [tex]\( 10500\% \)[/tex].
3. For [tex]\( b = 0.84 \)[/tex]:
[tex]\[
\text{Growth Rate} = (0.84 - 1) \times 100 = -0.16 \times 100 = -16\%
\][/tex]
The weekly percent growth rate is [tex]\( -16\% \)[/tex].
4. For [tex]\( b = 0.77 \)[/tex]:
[tex]\[
\text{Growth Rate} = (0.77 - 1) \times 100 = -0.23 \times 100 = -23\%
\][/tex]
The weekly percent growth rate is [tex]\( -23\% \)[/tex].
### Check Statements Against Calculations:
- Statement A: If [tex]\( b = 1.8 \)[/tex], the weekly percent growth rate of the share's value is [tex]\( 8\% \)[/tex].
- From our calculation, for [tex]\( b = 1.8 \)[/tex], the growth rate is [tex]\( 80\% \)[/tex]. This does not match the statement.
- Statement B: If [tex]\( b = 106 \)[/tex], the weekly percent growth rate of the share's value is [tex]\( 6\% \)[/tex].
- From our calculation, for [tex]\( b = 106 \)[/tex], the growth rate is [tex]\( 10500\% \)[/tex]. This does not match the statement.
- Statement C: If [tex]\( b = 0.84 \)[/tex], the weekly percent growth rate of the share's value is [tex]\( 84\% \)[/tex].
- From our calculation, for [tex]\( b = 0.84 \)[/tex], the growth rate is [tex]\( -16\% \)[/tex]. This does not match the statement.
- Statement D: If [tex]\( b = 0.77 \)[/tex], the weekly percent growth rate of the share's value is [tex]\( 23\% \)[/tex].
- From our calculation, for [tex]\( b = 0.77 \)[/tex], the growth rate is [tex]\( -23\% \)[/tex]. This matches the statement.
### Conclusion:
Among the given statements, the correct one is:
D. If [tex]\( b = 0.77 \)[/tex], the weekly percent growth rate of the share's value is [tex]\( -23\% \)[/tex].
This matches both the provided base value and the calculated growth rate. Therefore, the correct statement is D.
