Question Video: Understanding Exponential Decay | Nagwa Question Video: Understanding Exponential Decay | Nagwa

# Question Video: Understanding Exponential Decay Mathematics • Higher Education

At the start of an experiment, a scientist has a sample that contains 250 milligrams of a radioactive isotope. 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. b) Find the half-life of the isotope, giving your answer to the nearest minute.

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### Video Transcript

At the start of an experiment, a scientist has a sample which contains 250 milligrams of a radioactive isotope. 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. And b) Find the half-life of the isotope, giving your answer to the nearest minute.

Remember, we can model exponential growth and decay using the formula π of π‘ equals π nought times π to the power of ππ‘, where π nought is the initial value of π, and π is the rate of growth or decay. In our case, there are 250 milligrams of the sample at the start of the experiment. So, π nought must be equal to 250. We said that π is the rate of growth or decay. Since the isotope is decaying, this value is going to be negative. And 1.3 percent as a decimal is 0.013. So, π is negative 0.013. 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‘.

Weβre now going to look at part b). Here, it helps us to understand the definition of the phrase half-life. Itβs the time taken for the radioactivity of the isotope to fall to half of its original value. In our case, thatβs half of 250. Thatβs 125 milligrams. We need to work out for what value of π‘ π is equal to 125. We therefore set π equal to 125 and solve for π‘.

We see that 125 equals 250 times π to the power of negative 0.013π‘. We then divide both sides of this equation by 250. 125 divided by 250 is 0.5. So, we see that 0.5 equals π to the power of negative 0.013π‘. 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‘. But of course, the natural log of π to the power of negative 0.013π‘ is just negative 0.013π‘. And so, our final step is to divide both sides of this equation by negative 0.013. That gives us π‘ equals 53.319 minutes which, correct to the nearest minute, is 53 minutes.

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