Video: Applications of Exponential Growth

The number of cars worldwide, 𝑡 years after 2015, can be modeled by the formula 𝑁 = 10⁹ ⋅ 𝑒^(0.0277𝑡). In what year will there be 1.4 billion cars worldwide?

05:18

Video Transcript

The number of cars worldwide, 𝑡 years after 2015, can be modeled by the formula 𝑁 equals 10 to the power of nine multiplied by 𝑒 to the power of 0.0277𝑡. In what year will there be 1.4 billion cars worldwide?

Well, the first thing we need to look at when we’re solving this problem is 1.4 billion because we’re looking to find out in what year there will be 1.4 billion cars worldwide. Well, if we think about one billion, well, one billion is equal to one with nine zeros. It’s also 1000 million. So how can we also write this in a different way? Well, we could write it in standard form as one multiplied by 10 to the power of nine or as an exponent, which is 10 to the power of nine cause it both gives the same thing, which is one billion.

Well, therefore, as we know how to write one billion in other forms, we can look at writing 1.4 billion because if we want to get from one billion to 1.4 billion, what we do is multiply it by 1.4. So therefore, if we do the same with our standard form, what we’ll get is 1.4 multiplied by 10 to the power of nine. That’s another way that we can write 1.4 billion. So now, as we’ve got 1.4 billion in standard form, what we’re going to do is look at what the question is asking for. And the question is asking us to find out what year there will be 1.4 billion cars worldwide.

So it’s now that we can use our formula 𝑁 is equal to 10 to the power of nine multiplied by 𝑒 to the power of 0.0277𝑡. Well, the first thing we know is that the 𝑁 value is gonna be equal to 1.4 multiplied by 10 to the power of nine. And that’s because 𝑁 is the number of cars worldwide. So therefore, the number of cars worldwide that we’re looking at is 1.4 billion. So now, if we substitute our 𝑁 value into our formula, we’re gonna get 1.4 multiplied by 10 to the power of nine is equal to 10 to the power of nine multiplied by 𝑒 to the power of 0.0277𝑡. So now, what we want to do is to solve to find 𝑡.

And the first thing we’re gonna do is divide each side of the equation by 10 to the power of nine. And when we do that, the 10 to the power of nine on each side of the equation is gonna cancel out. So we’re gonna be left with 1.4 is equal to 𝑒 to the power of 0.0277𝑡. So now to solve the equation to find 𝑡, what we’re gonna do is take the natural logarithm of both sides of our equation. Well, why do we do that? We do that because if we take the natural logarithm of 𝑒 to the power of 𝑥, we’re just left with 𝑥, which is gonna be really useful.

So if we apply it to our equation and we take the natural logarithm of both sides, we’re gonna get ln or the natural logarithm of 1.4 is equal to 0.0277𝑡. And that’s because, as we know, we said if we take the natural logarithm of 𝑒 to the power of anything, then the answer is just that anything. And in our case, the anything was the 0.0277𝑡. So now, as we’re looking to solve to find 𝑡, we’re gonna do the inverse operation. Because we’ve got 0.0277 multiplied by 𝑡, we’re gonna divide each side of the equation by 0.0277. And when we do that, we’re gonna get ln 1.4 divided by 0.0277 is equal to 𝑡. Well, when we plug this into our calculator to find out what 𝑡 is, we get 𝑡 is equal to 12.147 et cetera.

So great! Does this mean that we’ve solved the problem? Well, no, because what we’ve been asked to find is in what year there will be 1.4 billion cars worldwide. And what we know so far is that after 12.147 years, there will be 1.4 billion cars worldwide. But what year will this be in? Well, our starting point was 2015. And we know that we’re gonna hit 1.4 billion cars worldwide 12.147 years after 2015. So therefore, we know that it’s gonna be in the year that is 12 years after 2015. So what we need to do is add 12 onto 2015. So this gives us an answer of 2027.

So therefore, we can say that if the number of cars worldwide, 𝑡 years after 2015, can be modeled by the formula 𝑁 equals 10 to the power of power of nine multiplied by 𝑒 to the power of 0.0277𝑡. Then the year in which there will be 1.4 billion cars worldwide is 2027. And we worked this out because we found out that there will be 1.4 billion cars worldwide 12.147 years after 2015. So therefore, we know there’s going to be 1.4 billion cars worldwide somewhere between 2027 and 2028. So therefore, it’s gonna fall in the year 2027.

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