Question

A candle is 17 in. tall after burning for 3 hours. After 5 hours, it is 15in tall. Write a linear equation to model the relationship between height h of the candle and time t. Predict how tall the candle will be after burning 8 hours.

**WILL GIVE MEDAL TO #1 HELPER**

Answer 1

@zepdrix
I know you've done this one before! :D

Answer 2

Oh man the stupid candle question again? +_+ haha

Answer 3

sec, brb

Answer 4

ok ok ok ok ok ok something about candle?

Answer 5

Yes. im confused on the formula? if there even is one.

Answer 6

So they told us that the relationship can be modeled by a `linear equation`.
A linear equation is of the form:\[\Large y=mx+b\]Where m is the slope of the line,
and b is the y-intercept.

Answer 7

Since we're dealing with height and time, instead of the normal x and y, let's plug in different variables.

Hopefully this won't get too confusing for you :)\[\Large h=mt+b\]

Answer 8

The information they gave us about the height at specific times we can think of as coordinate pairs.\[\Large (t_1,\;h_1)\quad=\quad (3,\;17)\]

Answer 9

\[\Large (t_2,\;h_2)\quad=\quad (5,\;15)\]

Answer 10

We can use these coordinate pairs to find the slope of our line.
Remember how to find slope? :)

Answer 11

Slope isn't y = mx + b, is it?

Answer 12

So slope is m, m = (y_f - y_i) / (x_f - x_i)

Answer 13

slope is\[\Large m=\frac{y_2-y_1}{x_2-x_1}\]

Answer 14

Oh, right right right.

Answer 15

Change in y, over change in x :D
we're using h and t though,
so our setup will look like:\[\Large m\quad=\quad\frac{h_2-h_1}{t_2-t_1}\]

Answer 16

Alright, Im on top of things.
What next?

Answer 17

Do I just plug-in now?

Answer 18

Plug in the m, yes.

Answer 19

What'd we end up with? :)

Answer 20

One sec.

Answer 21

uuuh...I got 7/5 (1.4)?
17-3/15-5

Answer 22

Hmm your setup looks a little strange.\[\Large h_2=17, \qquad h_1=15\]So the top of our slope formula should look like 17-15

Answer 23

Oh, that makes more sense. okay one sec again

Answer 24

-1?

Answer 25

cool, sounds right :)

Answer 26

WOO! :D

Answer 27

So plugging in our m, we currently have:\[\Large h=-t+b\]

Answer 28

To find our y-intercept (b), we can simply plug in one of the coordinate pairs and solve for b.

Answer 29

How bout we useeeee, (5,15).

Answer 30

so we're doing h = -5 + 15? kinda confused.

Answer 31

They gave us these "coordinate pairs" to plug in for the `h` and `t` values in the equation.

Answer 32

\[\Large (\color{orangered}{t},\;\color{royalblue}{h})=(\color{orangered}{5},\color{royalblue}{15})\]We want to plug these values \[\Large \color{royalblue}{h}=-\color{orangered}{t}+b\]into our equation:

Answer 33

Blah that pasted a little funky lol

Answer 34

Lol, well soo...
15 = -5 + b and we're trying to figure out b.

Answer 35

Yah that sounds right.

Answer 36

So then what? do you do the thing where you switch b with 15 (basically) but + changes to -?

Answer 37

Hmmm, no.
You're over thinking this I guess. :(

15 = -5 + b
T solve for b, add 5 to each side.

Answer 38

So doing so, it would be 20 = b since -5 cancels out...?

Answer 39

b = 20??

Answer 40

yup!

Answer 41

YESS!!

Answer 42

So that finishes the first part that we needed to do.
`Finding a linear equation for our relationship`.\[\Large h=-t+20\]

Answer 43

Do you understand what the 20 represents?
That is the initial h value.

So when t = 0, h = 20.
Hmm, what does that mean?

Answer 44

20 = 0 + 20...?

Answer 45

No, i mean.. what does that number `represent` ?

Answer 46

oh, lol XD it can either be time or height...and it has to be height because 20 represents the initial h value, right?

Answer 47

Ya :o Initial height, right?
So it's telling us that the candle was 20 inches long before it started burning.

Answer 48

Just an interesting little tid bit there +_+ ya.

Answer 49

It helps though! :D

Answer 50

Now that you've got your linear equation set up, they want you to find `h` when `t=8`.

Answer 51

h = -8 + 20?

Answer 52

(which is 12 btw)

Answer 53

yay team \c:/ looks good.

Answer 54

That's it?? :D

Answer 55

mhm

Answer 56

Thank you so much! I thought I was dead in water there for a moment :DDDDD