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Linear and exponential growth — Harder example

Watch Sal work through a harder Linear and exponential growth problem.

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  • blobby green style avatar for user kho102
    At , how did you get 1.0241? I don't understand where or how you got the 1.
    (35 votes)
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    • mr pants teal style avatar for user Hanna S.
      The equation for exponential growth is y=a(1+r)^t, with "a" as the initial amount, "r" as the growth rate (typically a percentage), and "t" as the amount of time intervals that have elapsed. Since the corn yield grows by 2.41% each year, you convert 2.41% into decimal form, which is 0.0241. Adding 1 to 0.0241 equals 1.0241, which is the number that Sal placed into his equation.

      He got the 1 because if you just multiplied 0.0241, your y-values would decrease because you're solving for 2.41% of the initial amount. Thus, you have to add the 1 and look for 102.41% of the initial amount in order to get the correct answer.
      (77 votes)
  • starky tree style avatar for user Mina Van
    At , why did he put a 1 on 1.0241?
    (8 votes)
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    • aqualine ultimate style avatar for user Axel
      This is because the corn grows by 2.41% each year. So since the original amount of corn is 100%, you have to add the original amount of corn to the percentage of its growth, 100%+2.41%, the result is 1.0241, and then you multiply by this amount to calculate the actual amount of corn growth,
      (33 votes)
  • male robot hal style avatar for user Rayhan Nirjhor
    What Sal did at created a slight bit of confusion for some people. Here is another simpler version (according to me).
    'Corn yield grew by approximately 2.41% per year'. So 2.41% of 51.2= 1.23392.
    So at 1st year= 51.2+1.23392
    at 2nd year= (51.2+1.23392)+1.23392
    [1.23392 increase of 1st year]
    at 3rd year= {(51.2+1.23392)+1.23392}+1.23392
    [1.23392 increase of the 2nd year]

    So at 15th year= 51.2+1.23392^15=74.60
    Therefore: 74.60-28.75=45.85
    (9 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      Well, unfortunately your alternate approach contains a couple of confusions. In this case you lucked out, but you still had to pick the closest answer, and if the rate of increase had been a little larger, your estimate would have pointed the way to an incorrect answer. What if they had offered both 44 and 46 as possible answers?
      Your answer, while close, would have rounded to the wrong choice.

      The reason is that after the first year where you multiplied by 2.41%, you have added the same amount (2.41% times the beginning amount) every year.
      As you have said, at 3rd year= {(51.2+1.23392)+1.23392}+1.23392
      this is actually veering off the correct answer, which is 54.992, while your answer is 54.902
      What you are showing is a linear functions of 51.2 + 1.23392(t) rather than the correct exponential function that you would need for a safe answer.
      But then you summarized this as
      So at 15th year= 51.2+1.23392^15=74.60
      But what you showed was 51.2 +1.23392*15 which gives only 69.709, which is quite a ways off.
      The tipoff to how to solve this is that the question says that the amount grew by 2.41% EACH year, not by 2.41% of the first year's amount added every year. The total difference in the final year is not a large amount; however, it is enough to cause you to usually miss this kind of question on the SAT.
      So, instead, to avoid being confused yourself on the SAT, you may want to practice building exponential functions--they are very common on the SAT, ACT and about any other test you can think of.

      The skeleton building block for exponential functions is
      ORIGINAL amount times (rate of increase or decrease) raised to the variable power
      In this case, it would be C(t) = 51.2(1.0241)ᵀ
      Here Sal showed this ready to solve as:
      Corn = 51.2(1.0241)¹⁵
      (18 votes)
  • marcimus orange style avatar for user soumya
    How can I practice more questions of same kind
    (4 votes)
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  • blobby green style avatar for user Elijah Johnston
    At how did he get 1.0241? I’m not exactly sure where the 1 came from?
    (4 votes)
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  • piceratops seedling style avatar for user Ramisha Rahman
    I dont understand where you got the 1.0241 from?
    (2 votes)
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  • blobby green style avatar for user Kevin.DaveJ
    How does 2.41% change to 1.0241?
    (1 vote)
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  • male robot johnny style avatar for user tejas.bv2001
    could you please tell me how did you get 1.0241 for 2.41 percentage
    (2 votes)
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    • male robot hal style avatar for user Rayhan Nirjhor
      What Sal did at created a slight bit of confusion for some people. Here is another simpler version (according to me).
      'Corn yield grew by approximately 2.41% per year'. So 2.41% of 51.2= 1.23392.
      So at 1st year= 51.2+1.23392
      at 2nd year= (51.2+1.23392)+1.23392
      [1.23392 increase of 1st year]
      at 3rd year= {(51.2+1.23392)+1.23392}+1.23392
      [1.23392 increase of the 2nd year]

      So at 15th year= 51.2+1.23392^15=74.60
      Therefore: 74.60-28.75=45.85
      (2 votes)
  • leaf orange style avatar for user krupari28
    At why did Sal add 23.5, 3.5 and 1.75?
    Where did the 3.5 and the 1.75 come from?
    (2 votes)
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    • mr pants purple style avatar for user Olivia  **INACTIVE**
      In 1959, the soybean yield was 23.5 bushels per acre (bu/A). According to the problem, the soybean yield grew by 3.5 bu/A every ten years. Since we want to figure out how much the soybean yield was in 1974 (fifteen years after 1959), we need to add 3.5 bu/A for every ten years that go by. So from 1959 to 1969 we add 3.5 bu/A to the original 23.5, getting us 27 bu/A. From 1969 to 1974 is only 5 years, which is half of 10 years, so we need to divide the rate of 3.5 bu/A for every 10 years by 2 to get the rate for every 5 years. 3.5 divided by 2 is 1.75, and we add that to the 27 bu/A to get 28.75.
      (2 votes)
  • leafers ultimate style avatar for user Mathew
    Why is it not 1.241, why did sal put a zero there.
    (2 votes)
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    • piceratops ultimate style avatar for user Simmon
      2.41 is a percent, so to find the actual number of bu/A in 2.41 percent of the total you would need to divide 2.41 by 100 first (percent=per 100 parts). To plug the percent into the growth formula you need to add 1 to account for the initial value, so you get 1+(2.41/100)=1.0241.
      (2 votes)

Video transcript

- [Instructor] We're told Sam needs to sign 300 copies of their new novel. They sign the copies at a constant rate. After 15 minutes, they have signed 20% of the copies. Which of the following equations models the number of copies of Sam's new novel, N, left assign T minutes after they started signing? So pause this video and see if you can have a go at this on your own. All right, now let's work through this together. So when I look at the choices, we have an exponential for choice A, exponential for choice B, and then we have two linear functions for choices C and D. Well, they tell us that they signed, Sam signs the copies at a constant rate. So if we're signing at a constant rate, an exponential is not going to describe either how many we've signed or how many are left to sign. So we can immediately rule out choices A and B. So to figure out between these last two choices, let's set up a little table here where we know that T is in minutes and N is a number of books left to sign. And they tell us after 15 minutes they have signed 20% of the copies. So after 15 minutes, what's end going to be? Well, N is the number of books left to sign. So that means that there's 80% of the books left to sign, 80% of the original number of books. 80% times 300 is going to be 240 books left to sign. So let's see which of these choices is consistent with that. So 300 minus four times 15. Let's see, 300 minus four times 15. That does indeed look like 240. So this one's looking good. What about 300 minus 20 times 15? Well, 20 times 15 is 300. So that means that N would be zero here. We know that Sam doesn't have, isn't done signing after 15 minutes. So we could rule this choice out and we like choice C.