Answer:
f(t) = 250[tex]e^{-0.007752t}[/tex]
Step-by-step explanation:
Let f(t) = [tex]\alpha[/tex][tex]e^{\beta t }[/tex]
where f is the amount of radioactive substance in grams
and t is the time in minutes
initially (at t=0), f = 250 grams
⇒f(0) = 250 grams
⇒[tex]\alpha[/tex][tex]e^{0\beta}[/tex] = 250
⇒[tex]\alpha[/tex][tex]e^{0}[/tex] = 250
⇒[tex]\alpha[/tex] = 250 grams {∵[tex]e^{0} = 1[/tex]}
⇒f(t) = 250[tex]e^{\beta t }[/tex]
At t = 250 minutes, f = 36 grams
⇒f(250) = 36 grams
⇒250[tex]e^{250\beta}[/tex] = 36
⇒[tex]e^{250\beta}[/tex] = [tex]\frac{36}{250}[/tex] = 0.144
⇒250[tex]\beta[/tex] = ㏑ 0.144 = -1.938
⇒[tex]\beta[/tex] = -[tex]\frac{1.938}{250}[/tex] = -0.007752 [tex]min^{-1}[/tex]
∴f(t) = 250[tex]e^{-0.007752t}[/tex]
