SOLUTION: Business and finance. In planning for a new item, a manufacturer assumes that the number of items produced x and the cost in dollars C of producing these items are related by a l

Algebra ->  Linear-equations -> SOLUTION: Business and finance. In planning for a new item, a manufacturer assumes that the number of items produced x and the cost in dollars C of producing these items are related by a l      Log On


   

Question 120653: Business and finance. In planning for a new item, a manufacturer assumes that
the number 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=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=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: is the first point (100,10000) and 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 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

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



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

C%28x%29=60x%2B4000 (note: C is much easier to remember as the Cost. Also notice how the equation is now in function notation)