Answer :
To convert the equation [tex]\( f(t) = 122(0.89)^t \)[/tex] to the form [tex]\( f(t) = a e^{k t} \)[/tex], we can follow these steps:
1. Identify the base of the exponential: In the given equation, the base of the exponent is 0.89.
2. Express the base in terms of the natural exponential function: We can use the property that any positive number can be expressed with base [tex]\( e \)[/tex] (the natural exponential number, approximately 2.718). To do this, we use natural logarithms:
[tex]\[
0.89 = e^{\ln(0.89)}
\][/tex]
Therefore, [tex]\( (0.89)^t = \left(e^{\ln(0.89)}\right)^t \)[/tex].
3. Apply the power rule of exponents: Using the property [tex]\((e^x)^y = e^{xy}\)[/tex], we get:
[tex]\[
(0.89)^t = e^{t \cdot \ln(0.89)}
\][/tex]
4. Substitute back into the equation: Replace [tex]\( (0.89)^t \)[/tex] with [tex]\( e^{t \cdot \ln(0.89)} \)[/tex] in the original equation:
[tex]\[
f(t) = 122 \times e^{t \cdot \ln(0.89)}
\][/tex]
Now, the equation is in the desired form [tex]\( f(t) = a e^{k t} \)[/tex], where:
- The coefficient [tex]\( a \)[/tex] is 122.
- The exponent's coefficient [tex]\( k \)[/tex] is the value of [tex]\( \ln(0.89) \)[/tex].
The calculated value of [tex]\( k \)[/tex] is approximately [tex]\(-0.11653381625595151\)[/tex].
So, in the form [tex]\( f(t) = a e^{k t} \)[/tex], we have:
- [tex]\( a = 122 \)[/tex]
- [tex]\( k \approx -0.11653381625595151 \)[/tex]
This completes the conversion of the equation to the requested form.
1. Identify the base of the exponential: In the given equation, the base of the exponent is 0.89.
2. Express the base in terms of the natural exponential function: We can use the property that any positive number can be expressed with base [tex]\( e \)[/tex] (the natural exponential number, approximately 2.718). To do this, we use natural logarithms:
[tex]\[
0.89 = e^{\ln(0.89)}
\][/tex]
Therefore, [tex]\( (0.89)^t = \left(e^{\ln(0.89)}\right)^t \)[/tex].
3. Apply the power rule of exponents: Using the property [tex]\((e^x)^y = e^{xy}\)[/tex], we get:
[tex]\[
(0.89)^t = e^{t \cdot \ln(0.89)}
\][/tex]
4. Substitute back into the equation: Replace [tex]\( (0.89)^t \)[/tex] with [tex]\( e^{t \cdot \ln(0.89)} \)[/tex] in the original equation:
[tex]\[
f(t) = 122 \times e^{t \cdot \ln(0.89)}
\][/tex]
Now, the equation is in the desired form [tex]\( f(t) = a e^{k t} \)[/tex], where:
- The coefficient [tex]\( a \)[/tex] is 122.
- The exponent's coefficient [tex]\( k \)[/tex] is the value of [tex]\( \ln(0.89) \)[/tex].
The calculated value of [tex]\( k \)[/tex] is approximately [tex]\(-0.11653381625595151\)[/tex].
So, in the form [tex]\( f(t) = a e^{k t} \)[/tex], we have:
- [tex]\( a = 122 \)[/tex]
- [tex]\( k \approx -0.11653381625595151 \)[/tex]
This completes the conversion of the equation to the requested form.
