Question

the 1981 explosion at a nuclear lab sent about 1000 kilograms of a radioactive element into the atmosphere. the function ( f(x)=1000(0.5)^{\frac{x}{30}} ) describes the amount, ( f(x) ), in kilograms, of a radioactive element remaining in the area ( x ) years after 1981. if even 100 kilograms of the radioactive element remains in the atmosphere, the area is considered unsafe for human habitation. find ( f(40) ) and determine if the area will be safe for human habitation by 2021.

( f(40)approx square ) (type an integer or a decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Substitute \(x = 40\) into the function

We have the function \(f(x)=1000(0.5)^{\frac{x}{30}}\). Substitute \(x = 40\) into it: \(f(40)=1000(0.5)^{\frac{40}{30}}\).

Step2: Simplify the exponent

\(\frac{40}{30}=\frac{4}{3}\). So \(f(40)=1000\times(0.5)^{\frac{4}{3}}\).

Step3: Calculate \((0.5)^{\frac{4}{3}}\)

\((0.5)^{\frac{4}{3}}=\sqrt[3]{0.5^{4}}=\sqrt[3]{\frac{1}{16}}\approx0.4\).

Step4: Calculate \(f(40)\)

\(f(40)=1000\times0.4 = 400.0\).

Answer:

\(400.0\)