Question Video: Solving Real-World Problems on Exponential Growth Modeled with Base e Mathematics

Video Transcript

When caffeine is metabolized by our body (that is, when our body breaks down, uses, and absorbs caffeine), the decreasing quantity of caffeine can be modelled by the function 𝑄 equals 𝑄 zero 𝑒 to the power of negative 0.1𝑡, where 𝑡 is the number of hours after an intake of 𝑄 zero. What is the half-life of caffeine in our body? In other words, how long does it take for our body to break down half of the caffeine? Round your answer to the nearest hour.

So in this problem, we have our function, which is 𝑄 is equal to 𝑄 zero 𝑒 to the power of negative 0.1𝑡. And we can say that 𝑄 is going to be equal to a half of 𝑄 zero. And that’s because it’s the half-life of caffeine that we’re looking for. So we want half of the caffeine that we started with to be broken down.

So therefore, if we want to represent this using our function, what we can say is that half the initial amount of caffeine is equal to the initial amount of caffeine multiplied by 𝑒 to the power of negative 0.1𝑡. So therefore, if we divide both sides of the equation by 𝑄 zero, we’re gonna get a half is equal to 𝑒 to the power of negative 0.1𝑡.

So to solve this, because we’ve got 𝑒 and we got 𝑒 to the power of negative 0.1𝑡 and we want to find out what 𝑡 is, we want to find the time, then we can take the natural logarithm of both sides of the equation. And when we do that, we’re gonna get the natural logarithm of a half is equal to the natural logarithm of 𝑒 to the power of 0.1𝑡. So then, we’re gonna get the natural logarithm of a half is equal to negative 0.1𝑡.

So now, if we flip it so that the 𝑡 is on the left-hand side of the equation and then we divide by negative 0.1, we get 𝑡 is equal to the natural logarithm of a half divided by negative 0.1. So this is gonna give us a value of 𝑡 of 6.9314, etc. Well, if we check out how the question wants us to leave our answer, it wants it rounded to the nearest hour. So therefore, we can say that the half-life of caffeine in our body is seven hours.

So what that means is it will take our body seven hours to break down half of the caffeine that we ingest.