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

`y=20e^{-(2)/(3)t}`

Exponential decay

Explanation:

We are given that

`(dy)/(dt)=-(2)/(3)y`

y=20 when t=0

`(dy)/(y)=-(2)/(3)dt`

Taking integration on both sides then we get

`lny=-(2)/(3)t+C`

Using formula
`\int (dx)/(x)=lnx,\int dx=x`

`y=e^{-(2)/(3)t+C}`

Using formula

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

`y=e^C\cdot e^{-(2)/(3)t}`

`e^C=Constant=C`

`y=Ce^{-(2)/(3)t}`

Substitute y=20 and t=0

`20=C`

Substitute the value of C

`y=20e^{-(2)/(3)t}`

When t tends to infinity then

`\lim_(t\rightarrow\infty)=\lim_(t\rightarrow\infty)20e^{-(2)/(3)t}=0`

When time increases then the value of function decrease

Hence, the function is exponential decay.