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

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) (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 and which results in the point (100,10000)

Also since 300 items will cost $22,000, this means and 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 (,) and (,)

Start with the slope formula (note:

is the first point (,) and

is the second point (,))

Plug in ,,, (these are the coordinates of given points)

Subtract the terms in the numerator to get . Subtract the terms in the denominator to get

Reduce

So the slope is

------------------------------------------------

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

------Point-Slope Formula------
where is the slope, and

is one of the given points

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

Plug in , , and (these values are given)

Distribute

Multiply and to get

Add to both sides to isolate y

Combine like terms and to get
------------------------------------------------------------------------------------------------------------
Answer:

So the equation of the line which goes through the points (,) and (,) is:

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

(note: C is much easier to remember as the Cost. Also notice how the equation is now in function notation)

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