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Answer :
To determine the rate of decay, [tex]\( r \)[/tex], for the given exponential function [tex]\( y = 63.4(0.92)^x \)[/tex], we need to understand how exponential functions express growth or decay.
In the function [tex]\( y = a(b)^x \)[/tex]:
- [tex]\( a \)[/tex] is the initial value (in this case, 63.4).
- [tex]\( b \)[/tex] is the base of the exponential function.
When [tex]\( b \)[/tex] is less than 1, the function represents exponential decay. The rate of decay is determined by how much less than 1 the base [tex]\( b \)[/tex] is.
For the function [tex]\( y = 63.4(0.92)^x \)[/tex]:
1. The base [tex]\( b \)[/tex] is 0.92.
2. Since 0.92 is less than 1, this indicates decay.
3. The rate of decay [tex]\( r \)[/tex] is found by calculating how much more is needed to reach 1, which is [tex]\( 1 - 0.92 \)[/tex].
Therefore, the rate of decay [tex]\( r \)[/tex] is:
[tex]\[ r = 1 - 0.92 = 0.08 \][/tex]
The correct answer is [tex]\( r = 0.08 \)[/tex].
In the function [tex]\( y = a(b)^x \)[/tex]:
- [tex]\( a \)[/tex] is the initial value (in this case, 63.4).
- [tex]\( b \)[/tex] is the base of the exponential function.
When [tex]\( b \)[/tex] is less than 1, the function represents exponential decay. The rate of decay is determined by how much less than 1 the base [tex]\( b \)[/tex] is.
For the function [tex]\( y = 63.4(0.92)^x \)[/tex]:
1. The base [tex]\( b \)[/tex] is 0.92.
2. Since 0.92 is less than 1, this indicates decay.
3. The rate of decay [tex]\( r \)[/tex] is found by calculating how much more is needed to reach 1, which is [tex]\( 1 - 0.92 \)[/tex].
Therefore, the rate of decay [tex]\( r \)[/tex] is:
[tex]\[ r = 1 - 0.92 = 0.08 \][/tex]
The correct answer is [tex]\( r = 0.08 \)[/tex].
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