Question

The 1984 explosion at a nuclear lab sent about 1000 kilograms of a radioactive element into the atmosphere. The function f(x)= 1000(0.5)^x/30 describes the​ amount, f(x), in​ kilograms, of a radioactive element remaining in the area x years after 1984. 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 2024.

Answer 1

By calculating f(40) for the given exponential decay function, we find that approximately 317 kilograms of the radioactive element would still be present in 2024. As this amount exceeds the 100 kilograms threshold, the area would remain unsafe for human habitation.

Step-by-step explanation:

The question involves using the exponential decay function f(x) = 1000(0.5)^(x/30) to determine the amount of a radioactive element remaining after a certain number of years.

The formula represents the decay of a radioactive substance over time, where x is the number of years after 1984, and f(x) is the mass of the radioactive substance remaining.

To find out if the area will be safe for human habitation by 2024, which is 40 years after 1984, we compute f(40). By plugging in 40 for x in the formula, we get:

f(40) = 1000(0.5)^(40/30)

f(40) = 1000(0.5)^(4/3)

f(40) = 1000(0.5)^(1.333...)

f(40) \approx 1000(0.317) \approx 317 kilograms

Since 317 kilograms is greater than the 100 kilograms threshold, the area would still be considered unsafe for human habitation in 2024.