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3. Factory in Omaha

Omaha's factory has yet another type of cost structure. It's costs function is provided graphically. It's maximum capacity is 38,000 units per day.

a. At what rate is the cost per unit decreasing for production levels above 12,000?

b. State the function for the domain [12000, 38000].

c. What is the cost per unit at the production level of 19,000?

Omaha's city council approved a special growth incentive that decreases the company's tax burden for production levels above last year's average. Explain how this is reflected by the Cost Function for Omaha's factory. ​

NO LINKS NO ASSESSMENTS OR TEST NOT MULTIPLE CHOICE3 Factory In OmahaOmahas Factory Has Yet Another Type Of Cost Structure Its Costs Function Is Provided Graphi class=

Sagot :

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.

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