Answers:
- a) cost is going down by 1 cent for every 1000 units produced
- b) f(x) = -0.00001x + 0.97
- c) $0.78 or 78 cents
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Explanation:
Part (a)
The two points (12000, 0.85) and (38000,0.59) of interest here. Use the slope formula on them
m = (y2-y1)/(x2-x1)
m = (0.59-0.85)/(38000-12000)
m = -0.26/26000
m = -0.00001
The rate is -0.00001 dollars per unit
In other words, the cost is going down by 0.00001 each time 1 unit is made.
This amount is incredibly small when focusing on one unit, but we can multiply by 10^3 (aka 1000) to get to -0.00001*10^3 = -0.01 which is more manageable.
What does this mean? It means that the cost is going down by 1 cent for every 1000 units produced, which to me seems like a more realistic phrase that is easier to grasp.
Of course, all of this applies only when 120000 < x < 38000.
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Part (b)
In the previous part, we found the slope was m = -0.00001
Plug in (x,y) = (12000, 0.85) and solve for b
y = mx+b
0.85 = (-0.00001)*(12000)+b
0.85 = -0.12+b
0.85+0.12 = b
b = 0.97
The y intercept is 0.97
The function for that diagonal line is f(x) = -0.00001x + 0.97
However, that function is restricted to the domain [12000, 38000] as the graph shows.
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Part (c)
Plug x = 19000 into the function we just found
Note how x = 19000 is in the domain [12000, 38000]
f(x) = -0.00001x + 0.97
f(19000) = -0.00001*19000 + 0.97
f(19000) = -0.19 + 0.97
f(19000) = 0.78
When you produce 19000 units, the cost per unit is $0.78 or 78 cents.
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As for the tax question, I'm not entirely sure. But it appears that the city council may have given the company tax relief as long as the company has made more than 12000 units (since this is where the cost curve starts to decrease). Perhaps this is the council's way of encouraging more economic activity and job growth.
Keep in mind that average cost curves tend to dip downhill when the amount produced goes up. After a certain point however, the average cost will start to increase. The company cannot expand forever and keep costs at a minimum. All of this is independent of taxes so it's not entirely clear if taxes played a role in that decreasing curve or not.
