Question

Determining Exponential Growth and Decay in Exercise, use the given information to write an exponential equation for y. Does the function represent exponential growth or exponential decay?
dy/dt = -2/3y, y = 20 when t = 0

Answer 1

Answer:

$y=20e^{-\frac{2}{3}t}$

Exponential decay

Step-by-step explanation:

We are given that

$\frac{dy}{dt}=-\frac{2}{3}y$

y=20 when t=0

$\frac{dy}{y}=-\frac{2}{3}dt$

Taking integration on both sides then we get

$lny=-\frac{2}{3}t+C$

Using formula  $\int \frac{dx}{x}=lnx,\int dx=x$

$y=e^{-\frac{2}{3}t+C}$

Using formula

$lnx=y\implies x=e^y$

$y=e^C\cdot e^{-\frac{2}{3}t}$

$e^C=Constant=C$

$y=Ce^{-\frac{2}{3}t}$

Substitute y=20 and t=0

$20=C$

Substitute the value of C

$y=20e^{-\frac{2}{3}t}$

When t tends to infinity then

$\lim_{t\rightarrow\infty}=\lim_{t\rightarrow\infty}20e^{-\frac{2}{3}t}=0$

When time increases then the value of function decrease

Hence, the  function is exponential decay.