A smoke detector contains 0.2 mg of Americium 241 (Am-241), a radioactive element that decays in t years according to the relation m = 0.2(0.5)^(t/432.2). Where m is the mass, in milligrams, remaining after t years.

A) The smoke detector will no longer work when the amount of Am-241 drops below half it's initial value. Is it likely to fail while you own it? Justify your answer.

B) If you buy a smoke detector today, how much of the Am-241 will remain after 50 years?

C) How long will it take for the amount of Am-241 to drop to 0.05 mg?

To answer these questions, we need to analyze the given equation m = 0.2(0.5)^(t/432.2), where m is the mass of Americium 241 remaining after t years.

A) The smoke detector will no longer work when the amount of Am-241 drops below half its initial value. We can set up an equation to find out if this happens while you own it:

0.5 * 0.2 = 0.2(0.5)^(t/432.2)

Simplifying the equation, we get:

0.1 = (0.5)^(t/432.2)

To solve for t, we can take the logarithm of both sides:

log(0.1) = log[(0.5)^(t/432.2)]

Using the logarithmic property, we can bring the exponent down:

log(0.1) = (t/432.2) * log(0.5)

Now, we can solve for t:

t = (432.2 * log(0.1)) / log(0.5)

Using a calculator, we find that t is approximately 1445.3 years.

Therefore, it is unlikely for the smoke detector to fail while you own it, considering that it takes around 1445 years for the amount of Am-241 to drop below half of its initial value.

B) To find out how much Am-241 will remain after 50 years, we can substitute t = 50 into the equation:

m = 0.2(0.5)^(50/432.2)

Calculating this expression, we get:

m ≈ 0.2(0.5)^(0.1157)

m ≈ 0.2 * 0.9737

m ≈ 0.1947 mg

Approximately 0.1947 mg of Am-241 will remain after 50 years.

C) To determine how long it will take for the amount of Am-241 to drop to 0.05 mg, we can set up the equation:

0.05 = 0.2(0.5)^(t/432.2)

Simplifying the equation, we have:

0.25 = (0.5)^(t/432.2)

Taking the logarithm of both sides, we get:

log(0.25) = log[(0.5)^(t/432.2)]

Using the logarithmic property, we can bring the exponent down:

log(0.25) = (t/432.2) * log(0.5)

Now, solving for t:

t = (432.2 * log(0.25)) / log(0.5)

Using a calculator, we find that t is approximately 288.8 years.

Therefore, it will take approximately 288.8 years for the amount of Am-241 to drop to 0.05 mg.