Question

The table below shows the fines imposed on a driver based on the rate of speed he or she was going when caught in a certain jurisdiction:
MPH : 60 65 70 80 85
FINE $: 80 105 130 180 205
1. Is the relationship between the rates of speed and the fines linear, quadratic, logarithmic, or exponential?
2. Based on this model, what fine would a driver receive if he or she were traveling 93 miles per hour? Show your equation as well as your answer.

Answer 1

Do you know how to find out if the points are linear or quadratic, etc?

Answer 2

No.... I don't really remember

Answer 3

The best way to see what the behavior of these coordinates does is to graph them on a graph. Let me do that and show you how I got it and what it looks like and then we will see from there, ok?

Answer 4

Okay thank you so much!

Answer 5

Ok, when you graph this, you get pretty much a straight line with a VERY steep slope. But it is a straight line, so a straight line is what type of relationship? Linear? Quadratic? Logarithmic? Exponential? Do you remember?

Answer 6

Linear?

Answer 7

Yes! Very good! That's almost half the battle. ; )

Answer 8

Now that you know that it is linear (all the points fall on a straight line) you can come up with an equation to fit that line and then find out what you want to know about 93 MPH. Ready?

Answer 9

Yeah

Answer 10

Ok. Now, I want to be able to explain this so it makes sense so I am going to insert some algebra lesson in here. Short but sweet. If you know the equation of a straight line, you can pick any point on the line and find out several things about the line. YOu can determine its y-intercept, its slope, the point-slope form of the equation and the slope-intercept form of the equation. With a formula for a line, you are able to use that formula (they are calling it a model) to determine other info or even predict things that might happen in the future as long as nothing about the linear relationships among the numbers changes. Ok, now I'm done being math teacher! ( I have to keep reminding myself that summer is here!)

Answer 11

We are going to use 2 of the coordinates that they have given us. Here , our x coordinate is the MPH and the y coordinate is the fine. Those are linear pairs of numbers. When x is 60, y = 80; when x = 65, y is 105; do you see that?

Answer 12

Yes

Answer 13

THey would look this if they were points you needed to graph.
(60, 80) (65, 105) (70, 130) (80, 180) (85, 205). And that is exactly what I did with them to find out that were a straight line. Now in order to find out what the fine would be if the driver were going 93 MPH, we have to find an equation. The fine is our y, and our unknown. The MPH is our x (93). Do you know the point-slope formula?

Answer 14

y-y1=m(x-x1) ??

Answer 15

Yes! very good again! But first, what do we need to find before we can actually use the point-slope formula?

Answer 16

The slope right?

Answer 17

Yes. Can you find the slope? Do you know how?

Answer 18

I don't remember... Is it the one where you take two points and minus them....

Answer 19

\[\frac{ y _{2}-y _{1} }{ x _{2}-x _{1} }\]That's the rise over run. Y is the rise because it's the vertical (up and down) axis, and x is the run because it's the side to side axis. Pick to points and fill them into the slope formula and tell me what you find to be the slope. Ok? Any two points from the data above. I picked (60, 80) and (80, 180). Just FYI.

Answer 20

100/20

Answer 21

So slope is 5

Answer 22

Yes again! And just so you know, if this is a straight line, the slope will ALWAYS come out to 5 no matter which points you choose to find the slope with. So yes, the slope is 5. Now, pick one of the points again and use it and the slope you just found to write an equation in point-slope form. Can you do that? Pick any point and use the x and the y from it and write an equation. Let me know if you struggle with that.

Answer 23

y-80=5(x-60) ???

Answer 24

Wow! Are you sure you need help?! Right again! I'm so glad you are so up on this stuff. The rest is a breeze!

Answer 25

Haha yeah I only remember certain things.

Answer 26

Ok, what you need to do now is to get that equation you wrote into standard form, which is y = mx + b. That means just distribute the 5 into the parenthesis and move the 80 over to the other side, away from the y. Got that?

Answer 27

y=5x-220 ?

Answer 28

Yes! You go girl!

Answer 29

Now, here's where you finally find the answer you are looking for. If they tell you to find the fine when the driver is going 93 MPH, what is your y and what is your x?

Answer 30

Would the y be 93 and then we find x?

Answer 31

Actually, if you look back at one of the notes I made, the one with all the sets of parenthesis and the coordinates in them, we designated x as the MPH and y as the fine. So what would be y and what would be x for us in the case of 93 MPH and its fine?

Answer 32

Oh so y=5(93)-220? Which means that the fine is $245?

Answer 33

OMG!!! You are so smart! Good for you! Yay! Give yourself an A+++

Answer 34

Haha thank you very much!