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Answer :
To find the monthly growth factor from the given function [tex]\( D(x) = 3125(1.65)^{\frac{x}{9}} \)[/tex], we'll follow these steps:
1. Understand the Current Function:
- The function models the average annual cost as an exponential function, where the base [tex]\( 1.65 \)[/tex] is the annual growth factor over a period of 9 years.
2. Relate Annual and Monthly Growth Factors:
- If we know the annual growth factor, we can find the monthly growth factor by taking the 12th root of the annual growth factor because there are 12 months in a year.
3. Calculate the Monthly Growth Factor:
- Given that the annual growth factor [tex]\( a \)[/tex] is [tex]\( 1.65 \)[/tex], the monthly growth factor [tex]\( m \)[/tex] can be calculated using the relation [tex]\( m^{12} = a \)[/tex], which gives us:
[tex]\[
m = a^{\frac{1}{12}} = 1.65^{\frac{1}{12}}
\][/tex]
4. Result:
- The calculated monthly growth factor is approximately [tex]\( 1.0426 \)[/tex].
5. Match with Given Options:
- We must convert this monthly growth factor to an expression that scales monthly growth over a period, represented as [tex]\( (monthly \, factor)^{12x} \)[/tex].
- Out of the given options, [tex]\( 3125(1.057)^{12x} \)[/tex] closely matches our calculated monthly growth factor.
Thus, the correct expression revealing the monthly growth factor is:
[tex]\[ 3125(1.057)^{12x} \][/tex]
1. Understand the Current Function:
- The function models the average annual cost as an exponential function, where the base [tex]\( 1.65 \)[/tex] is the annual growth factor over a period of 9 years.
2. Relate Annual and Monthly Growth Factors:
- If we know the annual growth factor, we can find the monthly growth factor by taking the 12th root of the annual growth factor because there are 12 months in a year.
3. Calculate the Monthly Growth Factor:
- Given that the annual growth factor [tex]\( a \)[/tex] is [tex]\( 1.65 \)[/tex], the monthly growth factor [tex]\( m \)[/tex] can be calculated using the relation [tex]\( m^{12} = a \)[/tex], which gives us:
[tex]\[
m = a^{\frac{1}{12}} = 1.65^{\frac{1}{12}}
\][/tex]
4. Result:
- The calculated monthly growth factor is approximately [tex]\( 1.0426 \)[/tex].
5. Match with Given Options:
- We must convert this monthly growth factor to an expression that scales monthly growth over a period, represented as [tex]\( (monthly \, factor)^{12x} \)[/tex].
- Out of the given options, [tex]\( 3125(1.057)^{12x} \)[/tex] closely matches our calculated monthly growth factor.
Thus, the correct expression revealing the monthly growth factor is:
[tex]\[ 3125(1.057)^{12x} \][/tex]
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