Question

The 1987 explosion at a nuclear lab sent about 1000 kilograms of a radioactive element into the atmosphere. the function f left parenthesis x right parenthesis equals 1000 left parenthesis 0.5 right parenthesis superscript startfraction x over 30 endfraction describes the​ amount, f(x), in​ kilograms, of a radioactive element remaining in the area x years after 1987. if even 100 kilograms of the radioactive element remains in the​ atmosphere, the area is considered unsafe for human habitation. find ​f(80​) and determine if the area will be safe for human habitation by 2067.

Answer 1

The equation tells you the half-life is 30 years. In 90 years, 3 half-lives will have passed, so the decay will be to (1/2)^3 = 1/8 its original level. Radiation will not have decayed to 1/10 its original level in 80 years. The area will be unsafe in 2067.

f(x) = 1000·0.5^(x/30)

f(80) = 1000·0.5^(80/30) ≈ 157.5 . . . . kilograms