Linear and exponential growth — Harder example
About this video
A constant signing rate produces a linear model, not an exponential one. Sam must sign 300 copies, and after 15 minutes 20 percent are signed, leaving 240 copies. The video rules out exponential choices for a constant rate and checks the linear options against that data point, which selects N = 300 - 4T.
You'll be able to
- recognize that a constant rate produces a linear model rather than an exponential one.
- convert a percentage into a number of items using the starting total.
- check candidate equations by substituting a known data point.
Key words
- Constant rate
- A change that happens by the same amount in each unit of time, which produces a linear relationship.
- Linear equation
- An equation whose graph is a straight line, written as a constant rate times time plus a starting value.
- Exponential model
- A function that changes by a constant percentage rather than a constant amount.
- Model check
- Substituting a known value into candidate equations to see which one matches the stated situation.
- Starting value
- The quantity at time zero, here 300 copies before signing begins.
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.
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