Question Video: Finding the Percentage Decrease Per Hour in a Model for Exponential Decay Mathematics

Video Transcript

The given figure shows the concentration 𝑐 in micrograms per liter of a certain drug in human blood plasma measured at different times. Considering that the concentration after ℎ hours can be modeled with the function 𝑐 is equal to 18 multiplied by 0.75 raised to the power ℎ, by what percentage does the drug’s concentration decrease every hour?

We’re given that the concentration of a particular drug in blood plasma can be modeled by the function 𝑐 is equal to 18 multiplied by 0.75 raised to the power ℎ. And that’s where ℎ is the time in hours. We’re asked to find the percentage decrease in concentration per hour.

To find this percentage, we first recall that an exponential decrease or decay can be modeled by the function 𝑐 of 𝑡 is 𝑎 multiplied by 𝑏 raised to the power 𝑡, where 𝑎 is the initial value and 𝑏 is between zero and one. We know also that this can be written equivalently as 𝑐 of 𝑡 is 𝑎 multiplied by one plus uppercase 𝑅 raised to the power 𝑡, where 𝑅 is the constant rate of change of 𝑐 and where, for exponential decay, 𝑅 is less than zero.

Now, we see that 𝑏 has been replaced by one plus 𝑅. And subtracting one from both sides, we find that 𝑏 minus one is equal to 𝑅. Now, in our equation for the concentration, comparing this to our general exponential decrease function, we find that 𝑎 is equal to 18 and that 𝑏 is equal to 0.75. And we know that our time variable is ℎ.

Now, recalling that the rate of change 𝑅 is equal to 𝑏 minus one, in our case that’s equal to 0.75 minus one, which is negative 0.25. This means that the drug’s concentration decreases at a proportionate rate of 0.25 per hour. And as a percentage, the rate of change is 100 multiplied by 𝑅, which in our case is 100 multiplied by negative 0.25. That’s minus 25 percent. Since this is negative, we say that the drug’s concentration decreases by 25 percent per hour.