Question

all exponential functions can be written in many forms. write the function $f(t) = 490e^{0.28t}$ in the form $f(t) = a(2)^{\frac{t}{k}}$. round all coefficients to four decimal places.

Explanation:

Step1: Set the two forms equal

We know that \( f(t) = 490e^{0.28t} \) and we want it in the form \( f(t)=a(2)^{\frac{t}{k}} \). So we set \( 490e^{0.28t}=a(2)^{\frac{t}{k}} \).

Step2: Rewrite the exponential with base 2

Recall that \( b^x = e^{x\ln b} \), so \( (2)^{\frac{t}{k}}=e^{\frac{t}{k}\ln 2} \). Then our equation becomes \( 490e^{0.28t}=a e^{\frac{t}{k}\ln 2} \).

Step3: Equate the coefficients and exponents

For the exponential functions to be equal, the coefficients must be equal and the exponents must be equal. So we have two equations:

1. Coefficient: \( a = 490 \) (since the coefficients of \( e^{\text{something}} \) must be equal)
2. Exponent: \( 0.28t=\frac{t}{k}\ln 2 \). We can divide both sides by \( t \) (assuming \( t

eq0 \)) to get \( 0.28=\frac{\ln 2}{k} \). Then we solve for \( k \): \( k = \frac{\ln 2}{0.28} \). Calculate \( \ln 2\approx0.6931 \), so \( k=\frac{0.6931}{0.28}\approx2.4754 \)

Step4: Write the function in the desired form

Now that we have \( a = 490 \) and \( k\approx2.4754 \), the function \( f(t) \) in the form \( a(2)^{\frac{t}{k}} \) is \( f(t)=490(2)^{\frac{t}{2.4754}} \)

Answer:

\( f(t) = 490.0000(2)^{\frac{t}{2.4754}} \) (or with more precise rounding during calculation, but following the steps above gives this result)