Answer :
Certainly! Let's solve this step-by-step:
The given exponential function is [tex]\( y = 63.4(0.92)^x \)[/tex].
In an exponential decay function of the form [tex]\( y = a(b)^x \)[/tex]:
- [tex]\( a \)[/tex] is the initial amount (63.4 in this case).
- [tex]\( b \)[/tex] is the base of the exponential, which tells us how much the quantity decreases as a power of [tex]\( x \)[/tex].
If [tex]\( b \)[/tex] is less than 1, it represents exponential decay, and the rate of decay, [tex]\( r \)[/tex], can be calculated as [tex]\( 1 - b \)[/tex].
Here's how the calculation works:
1. Identify the value of [tex]\( b \)[/tex] in your function. In this function, [tex]\( b = 0.92 \)[/tex].
2. Calculate the rate of decay [tex]\( r \)[/tex] using the formula:
[tex]\[
r = 1 - b
\][/tex]
Plug in the value of [tex]\( b \)[/tex]:
[tex]\[
r = 1 - 0.92
\][/tex]
3. Perform the subtraction:
[tex]\[
r = 0.08
\][/tex]
Therefore, the rate of decay expressed as a decimal is [tex]\( r = 0.08 \)[/tex]. This means each unit of time, the quantity decreases by 8%.
The given exponential function is [tex]\( y = 63.4(0.92)^x \)[/tex].
In an exponential decay function of the form [tex]\( y = a(b)^x \)[/tex]:
- [tex]\( a \)[/tex] is the initial amount (63.4 in this case).
- [tex]\( b \)[/tex] is the base of the exponential, which tells us how much the quantity decreases as a power of [tex]\( x \)[/tex].
If [tex]\( b \)[/tex] is less than 1, it represents exponential decay, and the rate of decay, [tex]\( r \)[/tex], can be calculated as [tex]\( 1 - b \)[/tex].
Here's how the calculation works:
1. Identify the value of [tex]\( b \)[/tex] in your function. In this function, [tex]\( b = 0.92 \)[/tex].
2. Calculate the rate of decay [tex]\( r \)[/tex] using the formula:
[tex]\[
r = 1 - b
\][/tex]
Plug in the value of [tex]\( b \)[/tex]:
[tex]\[
r = 1 - 0.92
\][/tex]
3. Perform the subtraction:
[tex]\[
r = 0.08
\][/tex]
Therefore, the rate of decay expressed as a decimal is [tex]\( r = 0.08 \)[/tex]. This means each unit of time, the quantity decreases by 8%.
