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 = -4y, y = 30 when t = 0

Answer 1

Answer:

$y=30e^{-4t}$

Exponential decay.

Step-by-step explanation:

We are given that

$\frac{dy}{dt}=-4y$

y=30 when t=0

Taking integration on both sides then we get

$\int \frac{dy}{y}=-4\int dt$

$lny=-4t+C$

By using the formula $\int \frac{dx}{x}=ln x,\int dx=x$

$y=e^{-4t+C}$

$y=e^{C}e^{-4t}=Ce^{-4t}$

Where$e^C=Constant=C$

$y=Ce^{-4t}$

Substitute y=30 and t=0

$30=C$

$y=30e^{-4t}$

Apply limit t tends to infinity

$\lim_{t\rightarrow \infty}=\lim_{t\rightarrow\infty}30e^{-4t}=0$

The value of function decreases with time therefore, it is an exponential decay.