Solved on Mar 14, 2024

Radioactive fallout from a 1981 nuclear lab explosion is described by f(x)=1000(0.5)x30f(x) = 1000(0.5)^{\frac{x}{30}}, where f(x)f(x) is the remaining amount (kg) xx years later. Determine f(40)f(40) and if the area will be safe by 2021.

STEP 1

1. The function f(x)=1000(0.5)x30f(x)=1000(0.5)^{\frac{x}{30}} accurately models the decay of the radioactive element over time.
2. The variable xx represents the number of years after 1981.
3. The area is unsafe for human habitation if there are 100 kilograms or more of the radioactive element remaining.
4. The year 2021 is 40 years after 1981.

STEP 2

1. Evaluate the function f(x)f(x) at x=40x=40 to find the amount of radioactive element remaining.
2. Compare the result of f(40)f(40) to the safety threshold of 100 kilograms to determine if the area is safe.

STEP 3

Substitute x=40x=40 into the function f(x)f(x) to calculate the amount of radioactive element remaining after 40 years.
f(40)=1000(0.5)4030 f(40) = 1000(0.5)^{\frac{40}{30}}

STEP 4

Simplify the exponent by dividing 40 by 30.
f(40)=1000(0.5)43 f(40) = 1000(0.5)^{\frac{4}{3}}

STEP 5

Calculate the value of 0.5430.5^{\frac{4}{3}}.
0.543=(0.53)4 0.5^{\frac{4}{3}} = \left(\sqrt[3]{0.5}\right)^4

STEP 6

Find the cube root of 0.50.5.
0.530.7937 \sqrt[3]{0.5} \approx 0.7937

STEP 7

Raise approximately 0.79370.7937 to the fourth power to get the value of 0.5430.5^{\frac{4}{3}}.
(0.7937)40.3969 (0.7937)^4 \approx 0.3969

STEP 8

Multiply 10001000 by the value obtained in the previous step to find f(40)f(40).
f(40)1000×0.3969 f(40) \approx 1000 \times 0.3969

STEP 9

Calculate the product to get the approximate amount of radioactive element remaining.
f(40)396.9 f(40) \approx 396.9

STEP 10

Compare the result of f(40)f(40) to the safety threshold of 100 kilograms.
Since f(40)396.9f(40) \approx 396.9 kilograms, which is greater than 100 kilograms, the area is still considered unsafe for human habitation by 2021.
The calculation shows that f(40)396.9f(40) \approx 396.9 kilograms of the radioactive element remains in the atmosphere by 2021, hence the area is not safe for human habitation.

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