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=(\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

`(19.75-36)/(10-0)=-1.625=-1.6\text{ grams/hour}`

Because the result is negative it is decreasing