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Shkiper50 [21]
3 years ago
6

A sample of a radioactive substance decayed to 97% of its original amount after a year. (Round your answers to two decimal place

s.) (a) What is the half-life of the substance? 22.76 Correct: Your answer is correct. yr (b) How long would it take the sample to decay to 85% of its original amount? yr
Mathematics
1 answer:
Andrej [43]3 years ago
8 0

Answer:

a) The half life of the substance is 22.76 years.

b) 5.34 years for the sample to decay to 85% of its original amount

Step-by-step explanation:

The amount of the radioactive substance after t years is modeled by the following equation:

P(t) = P(0)(1-r)^{t}

In which P(0) is the initial amount and r is the decay rate.

A sample of a radioactive substance decayed to 97% of its original amount after a year.

This means that:

P(1) = 0.97P(0)

Then

P(t) = P(0)(1-r)^{t}

0.97P(0) = P(0)(1-r)^{0}

1 - r = 0.97

So

P(t) = P(0)(0.97t)^{t}

(a) What is the half-life of the substance?

This is t for which P(t) = 0.5P(0). So

P(t) = P(0)(0.97t)^{t}

0.5P(0) = P(0)(0.97t)^{t}

(0.97)^{t} = 0.5

\log{(0.97)^{t}} = \log{0.5}

t\log{0.97} = \log{0.5}

t = \frac{\log{0.5}}{\log{0.97}}

t = 22.76

The half life of the substance is 22.76 years.

(b) How long would it take the sample to decay to 85% of its original amount?

This is t for which P(t) = 0.85P(0). So

P(t) = P(0)(0.97t)^{t}

0.85P(0) = P(0)(0.97t)^{t}

(0.97)^{t} = 0.85

\log{(0.97)^{t}} = \log{0.85}

t\log{0.97} = \log{0.85}

t = \frac{\log{0.85}}{\log{0.97}}

t = 5.34

5.34 years for the sample to decay to 85% of its original amount

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