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Current time:0:00Total duration:5:18

Linear and exponential growth — Harder example

Video transcript

- [Instructor] From 1925 to 2014, the United States corn yield, measured in bushels per acre, bushels per acre, grew by approximately 2.41% per year. By contrast, during the same time period, soybean yield grew by approximately 3.5 bushels per acre every 10 years. All right, so there's something interesting going on here. They've given us the, how much the corn yield grows in percentage terms, on a per year percentage terms, while for soybean yield, they tell us how much it grows in absolute terms, 3.5 bushels per acre every 10 years, or I guess you could say on average, 0.35, if you divide by 10, 0.35 bushels per acre per year. An NASS study, a NASS survey showed that in 1959, the corn yield was, actually I'm gonna make a little table here, I'm already sensing this'll be useful. So let's think about this. So we're gonna say, so an NASS survey showed that in 1959, the corn yield was 51.2 bushels per acre. So this is gonna be corn, and then they talk about soybean in a little bit, so soybean. So in 1959, in 1959, the corn yield was, so we're gonna assume everything is in bushels per acre. 51.2 bushels per acre, and the soybean yield was 23, 23.5 bushels per acre. Based on the information above, which of the following is the best estimate for the difference between the corn yield and the soybean yield in 1974? So 1974. So how many years have gone by? Let's see, this is going to be 15 years. So this is going to be plus 15 years from our 1959. So actually, the soybean is a little bit easier to calculate. 'Cause they just tell us the absolute number of bushels per acre, every 10 years. So 15 years, if you do 3 1/2 bushels per acre every 10 years, well 15 years is just gonna be half more of that again. So this is just going to be 23.5. So in 10 years, it'll grow by 3.5 bushels per acre, that's what they tell us, and then in another five years, it'll grow by half of this. Half of 10 years, well this is gonna grow by, let's see, half of 3.5 is what? It is 1.75. And so what is that going to be? Let's see, 23.5 plus 3.5 is, gets us to 27. And 27 plus 1.5 is going to be 28, or plus 1.75, is 28.75 bushels per acre in 1974. Did I do that right? 23.5 plus 3.5, is gonna be 27, 28.75, yep. So if we grow 3.5 bushels per acre in 10 years, we're gonna grow 1.75 bushels per acre in the next five years, that's the 15 years right over here. This is 10 years plus five years. So we get to 28.75. Now corn, what's corn going to be? Well here we're going by a percentage every year. So this is going to be 51.2, and we are growing, we're gonna multiply this times 1.0241. If you're growing by 2.41%, that's equivalent by, as multiplying by 1.0241. You grow, you keep what you have, and then you add 2.41%. Right over here. And you're gonna multiply that by as many years as you have. So it's going to be to the 15th power. Well I don't know what 1.0241 to the 15th power is, but luckily this is one of the questions where a calculator is encouraged. So let's figure this out. So let's figure out, let me make sure it's all cleared out. 1.0241 and let me raise it to the 15th power, is equal to 1.42, all of this business, and then multiply that times 51.2. So times 51.2, is equal to 73 point, well roughly 73.18. So this is approximately 73.18. Now what they want is, what's the best estimate for the difference between the corn yield and the soybean yield. So the difference, 73.18 minus 28.75, and they have a couple choices that are kinda close, so let me just make sure I do the math right. I could do this on paper, but we have our calculators. I'll just use that. So minus 28.75, to calculate the difference, gets us to about 44.4, and if we're gonna round to the nearest whole, that would be, or to the nearest whole, it would be 44. So it's this choice, that choice right over there.