Question

rewrite the exponential growth model using the variable k.
substitute \\( \frac { \ln 2 } { 40 } \\) for k.
\\( y = c e ^ { k t } \\)
model for exponential growth.
\\( y = \\)
substitute \\( c = 500 \\) and \\( k = \frac { \ln 2 } { 40 } \\).

Explanation:

Step1: Substitute \(C = 500\) and \(k=\frac{\ln 2}{40}\) into the model \(y = Ce^{kt}\)

$$y=500e^{\frac{\ln 2}{40}t}$$

Step2: Simplify the exponent using the property \(a\ln b=\ln(b^{a})\)

Since \(\frac{\ln 2}{40}t=\ln(2^{\frac{t}{40}})\), and \(e^{\ln x}=x\)
$$y = 500\times2^{\frac{t}{40}}$$

Answer:

\(y = 500e^{\frac{\ln 2}{40}t}\) or \(y = 500\times2^{\frac{t}{40}}\)