Question 159298
Let x=time elapsed (in hours) and y=height of candle (in inches)


Since the "candle is 8 inches tall after burning for 1 hour", this means that we have the values x=1 and y=8 which is the point (1,8)


Also, the statement "After 3 hours it is 5.5 inches tall" tells us that x=3 and y=5.5 which forms the point (3,5.5)



Now let's find the equation of the line through the points (1,8) and (3,5)




First let's find the slope of the line through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(3,5.5\right)]



{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(5.5-8)/(3-1)}}} Plug in {{{y[2]=5.5}}}, {{{y[1]=8}}}, {{{x[2]=3}}}, and {{{x[1]=1}}}



{{{m=(-2.5)/(3-1)}}} Subtract {{{8}}} from {{{5.5}}} to get {{{-2.5}}}



{{{m=(-2.5)/(2)}}} Subtract {{{1}}} from {{{3}}} to get {{{2}}}



{{{m=-1.25}}} Divide



So the slope of the line that goes through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(3,5\right)] is {{{m=-1.25}}}



Now let's use the point slope formula:



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-8=-1.25(x-1)}}} Plug in {{{m=-1.25}}}, {{{x[1]=1}}}, and {{{y[1]=8}}}



{{{y-8=-1.25x+(-1.25)(-1)}}} Distribute



{{{y-8=-1.25x+1.25}}} Multiply



{{{y=-1.25x+1.25+8}}} Add 8 to both sides. 



{{{y=-1.25x+9.25}}} Combine like terms. 




So the equation that goes through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(3,5.5\right)] is {{{y=-1.25x+9.25}}}



 Notice how the graph of {{{y=-1.25x+9.25}}} goes through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(3,5.5\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-1.25x+9.25),
 circle(1,8,0.08),
 circle(1,8,0.10),
 circle(1,8,0.12),
 circle(3,5.5,0.08),
 circle(3,5.5,0.10),
 circle(3,5.5,0.12)
 )}}} Graph of {{{y=-1.25x+9.25}}} through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(3,5.5\right)]

 



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Answer:


So the equation that models the height "y" at time "x" is 


{{{y=-1.25x+9.25}}}