SOLUTION: In planning for a new item, a manufacturer assumes that the numer of items produced x and the cost in dollars C of producing these items are related by a linear equation. Projectio

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Question 81319: In planning for a new item, a manufacturer assumes that the numer of items produced x and the cost in dollars C of producing these items are related by a linear equation. Projections are that 100 items will cost $10,000 to produce and that 300 items will cost $22,000 to produce. Find the equation that relates C and x.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=# of items and y=cost to produce
We essentially have 2 points (100,10000) and (300,22000) that we need to find the equation of a line that goes through those points

Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (100,10000) and (300,22000)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (100,10000) and (x%5B2%5D,y%5B2%5D) is the second point (300,22000))


m=%2822000-10000%29%2F%28300-100%29 Plug in y%5B2%5D=22000,y%5B1%5D=10000,x%5B2%5D=300,x%5B1%5D=100 (these are the coordinates of given points)


m=+12000%2F200 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%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and (x%5B1%5D,y%5B1%5D) is one of the given points


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


y-10000=%2860%29%28x-100%29 Plug in m=60, x%5B1%5D=100, and y%5B1%5D=10000 (these values are given)



y-10000=60x%2B%2860%29%28-100%29 Distribute 60


y-10000=60x-6000 Multiply 60 and -100 to get -6000%2F1. Now reduce -6000%2F1 to get -6000

y=60x-6000%2B10000 Add 10000 to both sides to isolate y


y=60x%2B4000 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%2B4000


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=60 and the y-intercept is b=4000


Notice if we graph the equation y=60x%2B4000 and plot the points (100,10000) and (300,22000), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=60x%2B4000 through the points (100,10000) and (300,22000)


Notice how the two points lie on the line. This graphically verifies our answer.





So the equation in terms of C and x is (just replace y with C)

C=60x%2B4000