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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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• At , how did you get 1.0241? I don't understand where or how you got the 1.
• 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.
• At , why did he put a 1 on 1.0241?
• 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,
• 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
• 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?

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
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
• How can I practice more questions of same kind
• Go online, and search up what you want to find.
(1 vote)
• At how did he get 1.0241? I’m not exactly sure where the 1 came from?
• 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.
(1 vote)
• I dont understand where you got the 1.0241 from?
• 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.
• How does 2.41% change to 1.0241?
(1 vote)
• 2.41/100=0.0241
But for the growth formula we must add 1 as in for accounting the initial value.
Hence it's 1+(2.41/100)=1+0.0241=1.0241
Hope you understood. :)
• could you please tell me how did you get 1.0241 for 2.41 percentage
• 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