Question

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.

Answer 1

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, $R=200x-\frac{x^2}{20}$
• Profit = R - C

$\frac{dx}{dt} =500$ When  Production output x=4000

Change in Cost

Cost, C=90000+40x

$\frac{dC}{dt} =40\frac{dx}{dt}=40*500\\\frac{dC}{dt}=20,000$

Change in Revenue

$\frac{dR}{dt} =200\frac{dx}{dt}-\frac{x}{10}\frac{dx}{dt}\\=200*500-\frac{4000}{10}*500\\\frac{dR}{dt} =-100,000$

Change in Profit

$\frac{dP}{dt} =\frac{dR}{dt}-\frac{dC}{dt}\\=-100,000-20,000\\\frac{dP}{dt}=-120,000$

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