An insurance company\'s monthly profit P(x, y), in thousand dollars, depends on

Question

An insurance company's monthly profit P(x, y), in thousand dollars, depends on the amount of money x (in thousands of dollars) spent on advertising per month and the number of sales associates y in its employ. Answer the following questions considering the given information: P(8, 25) = 93.5 partial differential p/partial differential x|_(8, 25) = 2.6 partial differential p/partial differential y|_(8, 25) = -1.2 Find the value of dy/dx|_(8, 25). Estimate the staffing change needed to compensate for a loss of $1, 800 in advertising per month if the agency still wished to bring in a monthly profit of $93, 500. Find the value of dx/dy|_(8, 25). Estimate the change in the advertising budget necessary to maintain a monthly profit of $93, 500 if the insurance hires 5 new sales associates.

Explanation / Answer

From the given question,

P(8,25)=$93.5 x 103

dP/dx=2.6

dP/dy=-1.2

a) dy/dx= (dP/dx)/(dP/dy)

=2.6/(-1.2)

=-2.167

b)P=(dP/dx)x +(dP/dy)y

93500=2.6x-1.2y

93500=(2.6)(x-1800)-1.2(y1)

93500=2.6x-4680-1.2*1000*y1

-1.2y=-4680-1.2*1000*y1

1.2*1000(y-y1)=4680

y-y1=3900/1000

Reduction in staff is 3900 /1000= 3.9 = 4 (approx.)

c) dx/dy=1/(dy/dx)

=1/(-2.137)

=-0.47

d)

P=(dP/dx)x +(dP/dy)y

93500=2.6x-1.2*1000y

93500=(2.6)(x1)-1.2*1000(y+5)

93500=2.6x1-1.2*1000*y-6000

2.6x-2.6x1=-6000

2.6(x-x1)=-6000

x-x1=2307

reduction in advertising is $ 2307

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