SOLUTION: I sent this one before but I have not received a response. Please help. Business and finance. In planning for a new item, a manufacturer assumes that the number of items produced

Algebra ->  Linear-equations -> SOLUTION: I sent this one before but I have not received a response. Please help. Business and finance. In planning for a new item, a manufacturer assumes that the number of items produced      Log On


   

Question 73868: I sent this one before but I have not received a response. Please help.
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!
If you were to plot these points, you would plot them on a graph where x is the number of items produced and y is the cost. If you found the slope of the line between the 2 points, you could find the equation. So lets find the slope:
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.



In other words, the rate of change of the cost to the number of items is $60 per unit. Now use the point-slope formula to find the equation
y-y%5B1%5D=m%28x-x%5B1%5D%29Plug in m=60 and to the equation
y-10000=60%28x-100%29
y-10000=60x-6000%29
y=60x-6000%2B10000%29
y=60x%2B4000%29
So this is your equation. If you plug in x=100 units, you should get y=$10,000 and if you plug in x=300, you should get y=$22,000 (y is the cost)


Check:
y=60%28100%29%2B4000%29
y=6000%2B4000%29
y=10000%29Works
y=60%28300%29%2B4000%29
y=18000%2B4000%29
y=22000%29Works
Hope this makes sense.