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Suppose that for a company manufacturing​ calculators, the​ cost, revenue, and profit equations are given by Start 1 By 3 Matrix 1st Row 1st Column Upper C equals 90 comma 000 plus 40 x comma Superscript 2nd Column Upper R equals 200 x minus StartFraction x squared Over 20 EndFraction comma 3rd Column Upper P equals Upper R minus Upper C Superscript EndMatrix where the production output in 1 week is x calculators. If production is increasing at a rate of 500 calculators per week when production output is 4 comma 000 calculators. Find the rate of increase​ (decrease) in​ cost, revenue, and profit.

Respuesta :

Answer:

  • Cost is Increasing at a rate of 20,000 per week
  • Revenue is decreasing at a rate of 100,000 per week
  • Profit is decreasing at a rate of 120,000 per week

Step-by-step explanation:

The cost, revenue, and profit equations for a company manufacturing​ calculators are given below:

  • Cost, C=90000+40x
  • Revenue, [tex]R=200x-\frac{x^2}{20}[/tex]
  • Profit = R - C

[tex]\frac{dx}{dt} =500\\[/tex] When  Production output x=4000

Change in Cost

Cost, C=90000+40x

[tex]\frac{dC}{dt} =40\frac{dx}{dt}=40*500\\\frac{dC}{dt}=20,000[/tex]

Change in Revenue

[tex]\frac{dR}{dt} =200\frac{dx}{dt}-\frac{x}{10}\frac{dx}{dt}\\=200*500-\frac{4000}{10}*500\\\frac{dR}{dt} =-100,000[/tex]

Change in Profit

[tex]\frac{dP}{dt} =\frac{dR}{dt}-\frac{dC}{dt}\\=-100,000-20,000\\\frac{dP}{dt}=-120,000[/tex]

Therefore, when production output is 4000 and increasing at a rate of 500 calculators per week,

  • Cost is Increasing at a rate of 20,000 per week
  • Revenue is decreasing at a rate of 100,000 per week
  • Profit is decreasing at a rate of 120,000 per week

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