Question

A radioactive substance is decaying according to the functionQ(t) = 36e^-0.061where Q(t) is the amount of the substance (in grams) and t is the time (inhours).a.) How many grams of the substance will there be after one day (t = 24hours)? Round to the nearest tenth.b.) How long will it take until only 1 gram of the substance is left? Round tothe nearest whole number.c.) how fast is the amount of the substance changing when t =10 hours round to the nearest tenth

Answer 1

We have the next function

$Q(t)=36e^{-0.06t}$

a)

For answer this section we need to find the value of Q when t=24

$Q(24)=36e^{-0.06(24)}=8.52=8.5\text{grams}$

b)

In order to know how long will take until 1 gram of substance left we need to isolate t of the formula and Q=1

$1=36e^{-0.06t}$

then we isolate t

$\begin{gathered} \ln (1)=\ln (36e^{-0.06t}) \\ \ln (1)=\ln (36)+\ln (e^{-0.06t}) \\ \ln (1)=\ln (36)+\ln (e^{-0.06t}) \\ \ln (1)-\ln (36)=-0.06t \\ t=\frac{\ln (1)-\ln (36)}{-0.06}=59.72=59.7\text{ hours} \end{gathered}$

c)

We have the initial amount that is when t=0

$Q(0)=36e^{-0.06(0)}=36$

when t=10

$Q(10)=36e^{-0.06(10)}=19.75$

Then we calculate

$\frac{19.75-36}{10-0}=-1.625=-1.6\text{ grams/hour}$

Because the result is negative it is decreasing