 APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS DECAY WORD PROBLEMS:

Problem 1: If you start a biology experiment with 5,000,000 cells and half the cells are dying every 10 minutes, how long will it take to have less than 1,000 cells?

Solution:

At time 0, there are 5,000,000 cells. At time 10 minutes, there are 0.50 x 5,000,000 = 2,500,000 cells remaining. At time 20 minutes, there are 0.50 x 2,500,000 = 1,250,000 cells remaining. At time 30 minutes, there are 0.50 x 1,250,000 = 625,000 cells remaining. When you plot the data, the curve looks exponential. Therefore, the mathematical model is probably exponential The model looks something like where represents the number of cells remaining after t minutes of observation, a represents the number of cells at the start of the experiment (5,000,000) , t represents the numbers of minutes since the experiment began, and b represents the decay constant based on a base of e.

We know that a=5,000,000 because we started with five million cells. However, you can verify it in the equation Let t=0 in the equation. The equation is now modified: We know that there are 2,500,000 cells after 10 minutes. Another way of saying this is that In the above equation, replace with 2,500,000 and replace t with 10.  Take the natural logarithm of both sides of the equation:   The equation describing the number of cells remaining after a certain number of minutes is Let's check it out by seeing if this model will give us 1,250,000 cells after twenty minutes. The model is How long will it take the sample to decay to below 1,000 cells? Just substitute 1,000 for in the equation. Take the natural logarithm of both sides of the equation.   It will take about 123 minutes for the cell population to drop below a 1,000 count.

If you would like to review problem 2, click on problem 2. [Exponential Rules] [Logarithms]

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