Answer :
To rewrite the function [tex]\( f(t) = 60000e^{0.15t} \)[/tex] in the form [tex]\( f(t) = a \cdot b^t \)[/tex], follow these steps:
1. Identify the initial coefficient: The initial coefficient [tex]\( a \)[/tex] is the same as the coefficient in front of the exponential function in the given equation. Therefore, [tex]\( a = 60000 \)[/tex].
2. Determine the base [tex]\( b \)[/tex]: In the expression [tex]\( e^{0.15t} \)[/tex], the base [tex]\( e \)[/tex] raised to the power of [tex]\( 0.15t \)[/tex] needs to be converted to a single base [tex]\( b \)[/tex] raised to [tex]\( t \)[/tex]. This is done by calculating [tex]\( b = e^{0.15} \)[/tex].
3. Calculate the value of [tex]\( b \)[/tex]: Consider [tex]\( e \approx 2.71828 \)[/tex]. By raising [tex]\( e \)[/tex] to the power of 0.15, you compute [tex]\( b \approx 1.1618 \)[/tex].
4. Round to four decimal places: Round both coefficients to four decimal places, which gives [tex]\( a = 60000 \)[/tex] and [tex]\( b = 1.1618 \)[/tex].
So, the function in the form [tex]\( f(t) = a \cdot b^t \)[/tex] is:
[tex]\[ f(t) = 60000 \cdot (1.1618)^t \][/tex]
These steps provide a clear way of converting the exponential form [tex]\( e^{kt} \)[/tex] to a base [tex]\( b \)[/tex] raised to the power of [tex]\( t \)[/tex], while maintaining the functionality of the equation.
1. Identify the initial coefficient: The initial coefficient [tex]\( a \)[/tex] is the same as the coefficient in front of the exponential function in the given equation. Therefore, [tex]\( a = 60000 \)[/tex].
2. Determine the base [tex]\( b \)[/tex]: In the expression [tex]\( e^{0.15t} \)[/tex], the base [tex]\( e \)[/tex] raised to the power of [tex]\( 0.15t \)[/tex] needs to be converted to a single base [tex]\( b \)[/tex] raised to [tex]\( t \)[/tex]. This is done by calculating [tex]\( b = e^{0.15} \)[/tex].
3. Calculate the value of [tex]\( b \)[/tex]: Consider [tex]\( e \approx 2.71828 \)[/tex]. By raising [tex]\( e \)[/tex] to the power of 0.15, you compute [tex]\( b \approx 1.1618 \)[/tex].
4. Round to four decimal places: Round both coefficients to four decimal places, which gives [tex]\( a = 60000 \)[/tex] and [tex]\( b = 1.1618 \)[/tex].
So, the function in the form [tex]\( f(t) = a \cdot b^t \)[/tex] is:
[tex]\[ f(t) = 60000 \cdot (1.1618)^t \][/tex]
These steps provide a clear way of converting the exponential form [tex]\( e^{kt} \)[/tex] to a base [tex]\( b \)[/tex] raised to the power of [tex]\( t \)[/tex], while maintaining the functionality of the equation.
