To identify the percent rate of change in the exponential function [tex]\( y = (0.9)^{\frac{t}{12}} \)[/tex], we need to analyze the form and properties of the given function.
The function [tex]\( y = (0.9)^{\frac{t}{12}} \)[/tex] is an example of an exponential function, where the base of the exponent is 0.9 and the exponent is a function of [tex]\( t \)[/tex]. In an exponential function of the form [tex]\( y = a^{bt} \)[/tex], the base [tex]\( a \)[/tex] tells us about the growth or decay process:
- If [tex]\( a \)[/tex] is greater than 1, the function represents exponential growth.
- If [tex]\( a \)[/tex] is between 0 and 1, the function represents exponential decay.
In this case, [tex]\( a = 0.9 \)[/tex], which is less than 1. Therefore, the function represents an exponential decay.
To find the percent rate of decay, we calculate [tex]\( (1 - a) \times 100\% \)[/tex]:
1. [tex]\( a = 0.9 \)[/tex]
2. Calculate [tex]\( 1 - 0.9 \)[/tex]:
[tex]\[ 1 - 0.9 = 0.1 \][/tex]
3. Convert the decay factor to a percentage by multiplying by 100:
[tex]\[ 0.1 \times 100\% = 10\% \][/tex]
Hence, the percent rate of decay for the function [tex]\( y = (0.9)^{\frac{t}{12}} \)[/tex] is [tex]\( 10\% \)[/tex].
So, the correct answer is:
[tex]\[ 10\% \text{ decay} \][/tex]
