Question 120653
Let x=items, y=cost to produce x amount of items


Since 100 items will cost $10,000, this means {{{x=100}}} and {{{y=10000}}} which results in the point (100,10000)



Also since 300 items will cost $22,000, this means {{{x=300}}} and {{{y=22000}}} which results in the point (300,22000)


So we have the two points: (100,10000) and (300,22000)


Let's find the equation of the line through these points


First lets find the slope through the points ({{{100}}},{{{10000}}}) and ({{{300}}},{{{22000}}})


{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula (note: *[Tex \Large \left(x_{1},y_{1}\right)] is the first point ({{{100}}},{{{10000}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{300}}},{{{22000}}}))


{{{m=(22000-10000)/(300-100)}}} Plug in {{{y[2]=22000}}},{{{y[1]=10000}}},{{{x[2]=300}}},{{{x[1]=100}}}  (these are the coordinates of given points)


{{{m= 12000/200}}} Subtract the terms in the numerator {{{22000-10000}}} to get {{{12000}}}.  Subtract the terms in the denominator {{{300-100}}} to get {{{200}}}

  


{{{m=60}}} Reduce

  

So the slope is

{{{m=60}}}


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Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
{{{y-y[1]=m(x-x[1])}}} where {{{m}}} is the slope, and *[Tex \Large \left(\textrm{x_{1},y_{1}}\right)] is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


{{{y-10000=(60)(x-100)}}} Plug in {{{m=60}}}, {{{x[1]=100}}}, and {{{y[1]=10000}}} (these values are given)



{{{y-10000=60x+(60)(-100)}}} Distribute {{{60}}}


{{{y-10000=60x-6000}}} Multiply {{{60}}} and {{{-100}}} to get {{{-6000}}}


{{{y=60x-6000+10000}}} Add {{{10000}}} to  both sides to isolate y


{{{y=60x+4000}}} Combine like terms {{{-6000}}} and {{{10000}}} to get {{{4000}}} 

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Answer:



So the equation of the line which goes through the points ({{{100}}},{{{10000}}}) and ({{{300}}},{{{22000}}})  is:{{{y=60x+4000}}}




Now simply replace y with C(x) to get the equation 


{{{C(x)=60x+4000}}} (note: C is much easier to remember as the <b>C</b>ost. Also notice how the equation is now in function notation)