Average Annual Public University costs The table shows the...

Average Annual Public University costs The table shows the cost of a year's tuition, room and board, and fees at 4 -year public colleges for the years $2006-2014$. Letting $y$ represent the cost in dollars and $x$ the number of years since $2006,$ the function $$f(x)=13,017(1.05)^{x}$$ models the data quite well. According to this function, in what year will the 2006 cost be doubled? (TABLE CANNOT COPY)

Average Annual Public University costs The table shows the cost of a year's tuition, room and board, and fees at 4 -year public colleges for the years $2006-2014$. Letting $y$ represent the cost in dollars and $x$ the number of years since $2006,$ the function
$$f(x)=13,017(1.05)^{x}$$
models the data quite well. According to this function, in what year will the 2006 cost be doubled?
(TABLE CANNOT COPY)

Key Concepts

Exponential Growth Function

This concept refers to a mathematical model in which a quantity increases by a constant percentage over equal time intervals. Such functions are commonly used to describe scenarios like tuition increases, population growth, and investment returns, where each period's growth builds upon the previous period's amount.

Doubling Time

Doubling time is the period it takes for a quantity following exponential growth to double in size. It is determined by setting the exponential model equal to twice its initial value and solving for the time variable, providing insight into how quickly the quantity is growing.

Logarithmic Equations

Logarithmic equations involve logarithms, which are the inverse operations of exponentiation. They play a crucial role in solving for the time variable in exponential growth models, as taking the logarithm of both sides of an equation allows the exponent to be isolated and calculated.

Transcript

00:01 All right, in this scenario, we're given a table of values of the costs of tuition, college tuition, for years after 2006.

00:11 So what i did was i made a table in desmos.

00:15 In 2006, i used as year zero since they said years after 2006.

00:22 Okay.

00:24 I also plugged in the function that they provided us in this problem that said represented the data well and we're going to look into that.

00:34 All right.

00:35 So first thing i would do is if you are going to graph this on desmos, i would adjust the windows.

00:41 So if you look in my graph setting, i did from 0 to 24 on the x -axis because it's years after 2006.

00:50 So i figured around 24 years would be sufficient for this problem.

00:55 And then i noticed that my lowest price point was 12 ,000 on here.

00:59 So i did from 10 ,000.

01:02 I'm actually going to change that to 26 ,000 because they want to know when that first year that 2006 is going to double.

01:11 All right.

01:12 So this is what the data point looks like on the graph.

01:18 Okay.

01:19 And then the function that they provided us that said, they said models it pretty well, looks like this.

01:26 So it does model it fairly well.

01:28 It goes through a lot of the points.

01:31 And we can use this model to help predict when this value is going to double.

01:37 All right.

01:38 So the reason i started it off in the graphing calculator is because i can show you guys that if you take this function here, the one that they give us to help us predict, you can actually, on desmos, hit the settings button and convert that to its own table.

01:56 Okay? so now what you can do is use this as a prediction.

02:01 Now their first year says that it was $13 ,000.

02:07 Our first year says it was $12 ,000 ,837.

02:10 It's pretty close.

02:11 All right.

02:12 We want to know when that $12 ,837 is going to double.

02:16 So we want to know when it gets to 25 ,674.

02:23 So the reason i suggest doing this is you can go to the two and just hit enter, and it will solve for each of the values for you...