The half-life of a certain radioactive substance is 20.5 years. You have 80 grams of the substance.
a. Find the decay rate, accurate to 5 decimal places.
b. Write the function that gives the amount that remains radioactive t years into the future.
c. Determine how many grams of the substance will remain radioactive 50 years from now.
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To solve this math problem, we can use the half-life formula:
Amount remaining = Initial amount * (1/2)^(time elapsed / half-life)
a. Find the decay rate:
The decay rate is the fraction of the substance that decays per unit of time. In this case, the half-life is 20.5 years. So the decay rate would be (1/2)^(1/20.5).
Using a calculator, we find that the decay rate is approximately 0.03348.
b. Write the function that gives the amount that remains radioactive t years into the future:
Let's call the function A(t), where t represents the time elapsed in years.
A(t) = 80 * (1/2)^(t / 20.5)
c. Determine how many grams of the substance will remain radioactive 50 years from now:
To find how many grams of the substance will remain radioactive 50 years from now, we substitute t = 50 into the function A(t) we derived in part b.
A(50) = 80 * (1/2)^(50 / 20.5)
Using a calculator, we find that A(50) is approximately 14.364 grams.
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