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### Course: Digital SAT Math>Unit 3

Lesson 7: Linear and exponential growth: foundations

# Linear and exponential growth — Harder example

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

## Want to join the conversation?

• 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.
• the way i did it, which will also help u in non mcq Qs is that 20% of 300 is 60, so he signed 60 copies in 15 mins, dived both the values and u get 4 copies per minute. therefore 300 - 4t
• This was easier than the basic one
• 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
• Why is the answer, not B?
• The key is in the question. It tells us that the copies were signed "at a constant rate". This means that the slope of our equation we'll make is going to be a constant, which means that it's just the equation of a line and not of exponential growth. If the question said something like "the rate at which copies were signed increased according to a common ratio" or anything really that mentions a ratio, you should be looking for exponential equations. Here though, we have a constant rate, which should allow us to automatically rule out A) and B).
• How can I practice more questions of same kind
• Very, very late answer but I think you could have used the Bluebook app
(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.
• How does 2.41% change to 1.0241?
• 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. :)
• Is this a problem that we are allowed to use a calculator on?

## 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.