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At the start of an experiment, a scientist has a sample which contains 250 milligrams of a radioactive isotope.
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The radioactive isotope decays exponentially at a rate of 1.3 percent per minute. a) Write the mass of the isotope in milligrams, π, as a function of the time in minutes, π‘, since the start of the experiment.
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And b) Find the half-life of the isotope, giving your answer to the nearest minute.
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Remember, we can model exponential growth and decay using the formula π of π‘ equals π nought times π to the power of ππ‘, where π nought is the initial value of π, and π is the rate of growth or decay.
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In our case, there are 250 milligrams of the sample at the start of the experiment.
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So, π nought must be equal to 250.
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We said that π is the rate of growth or decay.
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Since the isotope is decaying, this value is going to be negative.
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And 1.3 percent as a decimal is 0.013.
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So, π is negative 0.013.
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We can therefore say that the function that describes the mass in terms of minutes is π equals 250 times π the power of negative 0.013π‘.
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Weβre now going to look at part b).
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Here, it helps us to understand the definition of the phrase half-life.
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Itβs the time taken for the radioactivity of the isotope to fall to half of its original value.
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In our case, thatβs half of 250.
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Thatβs 125 milligrams.
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We need to work out for what value of π‘ π is equal to 125.
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We therefore set π equal to 125 and solve for π‘.
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We see that 125 equals 250 times π to the power of negative 0.013π‘.
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We then divide both sides of this equation by 250.
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125 divided by 250 is 0.5.
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So, we see that 0.5 equals π to the power of negative 0.013π‘.
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We then take the natural logarithm of both sides of this equation to get the natural log of 0.5 equals the natural log of π to the power of negative 0.013π‘.
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But of course, the natural log of π to the power of negative 0.013π‘ is just negative 0.013π‘.
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And so, our final step is to divide both sides of this equation by negative 0.013.
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That gives us π‘ equals 53.319 minutes which, correct to the nearest minute, is 53 minutes.