Interpreting time in exponential models | Mathematics II | High School Math | Khan Academy

After a special medicine is introduced into a petri dish full of bacteria, the number of bacteria remaining in the dish decreases rapidly. The relationship between the elapsed time T in seconds and the number of bacteria n of T in the petri dish is modeled by the following function.

Alright, complete the following sentence about the half-life of the bacterial culture: the number of bacteria is halved every blank seconds. So T is being given to us in seconds. So let's think about this a little bit. Let's think about a little table here. I'll draw a little table here.

So if we have this, this is T and this is n of T. I'll start with a straightforward T at time equals zero. Right when we start this whole thing, if this T is 0 then 1/2 to the 0 over 5.5 power, that's just 1 after the zero. That's all going to be 1, and you're just going to be left with one thousand bacteria in the petri dish.

Now, at what point do we get to multiply by 1/2? At what point do we get to say one thousand times 1/2? Well, in order to say one thousand times 1/2, the exponent here has to be one. So at what time is the exponent here going to be one? Well, the exponent here is going to be one this whole exponent's going to be one when T is equal to five point five seconds.

So T is five point five seconds, and likewise, we wait another five point five seconds. So if we go to eleven seconds, then this is going to be one thousand times eleven divided by five point five is two, so times 1/2 to the second power. So times 1/2 times 1/2.

So every five and a half seconds, we will essentially have half of the bacteria that we had five and a half seconds ago. So the number of bacteria is halved every five point five seconds, and you see it in the formula and in the function definition right over there. But it's nice to reason it through and really digest why it makes sense.

Let's do a few more of these. The chemical element einsteinium-253 naturally loses its mass over time. A sample of einsteinium-253 had an initial mass of 320 grams. When we measured it, the relationship between the elapsed time T in days and the mass M of T in grams left in the sample is modeled by the following function.

Complete the following sentence about the rate of change in the mass sample: the sample loses eighty-seven point five percent of its mass every blank days. So this one, instead of saying how much we grew or shrunk by, we're saying a percent change. So if you lose eighty-seven point five percent, that means that you are left with 12.5 percent, which is the same thing as saying that you have 0.125 percent of your mass.

So another way of thinking about it is the sample has now 0.125 of its original mass. So how long does it take for the sample to be 0.125 of its mass? And then we could do a similar idea. You see the 0.125 right over here, and so I could draw a table if you like. Although I think you might guess where this is going, but let me draw a little table here.

So T and M of T. When T is zero, M of T is three hundred twenty. At what point is T at what time is M of T going to be 320 times 0.125? Because this going from this to this is losing eighty-seven point five percent of your mass. Losing eighty-seven please, so let me just write it this way.

So this is minus 87.5 percent of the mass; you've lost 0.875 to get to 0.125. So this, you could just use 0.125 to the first power. So what T do you have to make this exponent equal one? Well, T has to be sixty-one point four. Sixty-one point four, and T is in days, sixty-one point four days.

Now, you might be tempted to always just pattern match, so whatever is in the denominator here. But I really encourage you to think about it because that's the whole point of these problems. If you just are pattern matching these, well, I don't know how helpful that's going to be for you.

Let's do one more of these. Howard started studying how the number of branches on his tree grows over time. The relationship between the elapsed time T in years since Howard started studying the tree and the number of its branches n of T is modeled by the following function.

Complete the following sentence about the rate of change in the number of branches: Howard's tree gains four-fifths more branches every blank years. So gaining four-fifths is equivalent to multiplying by... You're gaining four-fifths of what you already are. You're not just getting the number 4/5, you're getting 4/5 of what you already are.

So that's the equivalent of multiplying by one plus 4/5, or nine-fifths. So gaining four-fifths is the same thing as multiplying by nine-fifths. If I'm five years old and if I gain four-fifths of my age, I would gain five. I've gained four years to get to be nine years old, which means I've multiplied my age by nine-fifths.

So Howard's tree, you could say, grows by a factor of nine-fifths every how many years? Well, you could see over here the common ratio is nine-fifths. So you're going to grow by nine-fifths every time T is a multiple of seven point three. Or as you say, every time T increases by seven point three, you're going to... then this exponent is going to increase by a whole.

So you could view that as multiplying again by nine-fifths. So Howard's tree gains four-fifths more branches every seven point three years.