A chemical substance has a decay rate of $6.2 %$ per day. The rate of change of an amount $N$ of the chemical after $t$ days is given by $\frac{dN}{dt} = - 0.062 N$ .
a) Let $N_{0}$ represent the amount of the substance present at $t = 0$ . Find the exponential function that models the decay.
b) Suppose that $700 g$ of the substance is present at $t = 0$ . How much will remain after 5 days?
c) What is the rate of change of the amount of the substance after 5 days?
d) After how many days will half of the original $700 g$ of the substance remain?
a) $N (t) = N_{0} e^{- 0.062 t}$
b) After 5 days, $513 g$ will remain.
(Round to the nearest whole number as needed.)
c) After 5 days, the rate of change is $- 31.8 g /$ day.
(Round to one decimal place as needed.)
d) Half of the substance will remain after days.
(Round to one decimal place as needed.)

Answer

Expert–verified

Hide Steps

Answer

Final Answer: The exponential function that models the decay is $N (t) = N_{0} e^{- 0.062 t}$ .

Steps

Step 1 ：The rate of change of the amount of the substance is given by the differential equation $\frac{d N}{d t} = - 0.062 N$ . This is a first order linear differential equation. The general solution to this type of equation is given by $N (t) = N_{0} e^{k t}$ , where $N_{0}$ is the initial amount of the substance, $k$ is the rate of decay, and $t$ is the time. In this case, $k = - 0.062$ .

Step 2 ：So, the exponential function that models the decay is $N (t) = N_{0} e^{- 0.062 t}$ .

Step 3 ：Final Answer: The exponential function that models the decay is $N (t) = N_{0} e^{- 0.062 t}$ .

Answer

Popular Problems