Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

33 A manufacturer is planning to sell a new product at the price of $150 per unit and
estimates that if x thousand dollars is spent on development and y thousand

dollars is spent on promotion, approximately

320 160

24

y

yx

+

x

+ +

units of the product

will be sold. The cost of manufacturing the product is $50 per unit. If the

manufacturer has a total of $8,000 to spend on development and promotion, how

should this money be allocated to generate the largest possible profit? (Hint:

Profit=(number of units)(price per unit−cost per unit)−total amount spent on

development and promotion.)plitkrmñak;eRKaglk;plitplfμIkñúgtémø 150 duløakñúg1Ékta

ehIyrMBwgTukfaebIeKcMNayxBan;duløaeTAelIkarplit nigyBan;duløaeTAelIkarpSBVpSay enaHplit

plRbEhlCa

320 160

2

y

yx

+

4

x

+ +

nwgRtUvlk;dac;. éføedImelIkarplitKW 50 duløa. ebIplitkrman

8000 duløasRmab;cMNayelIkarplit nigpSBVpSay enaHetIeKKYrEbgEckdUcemþcedIm,I[R)ak;

cMeNjGtibrma? ¬ENnaM³ R)ak;cMeNj=¬cMnYnÉkta¦¬éfølk;kñúg1Ékta-éføcMNaykñúg1Ékta¦-

cMNaysrubelIkarplitnigpSBVpSay¦¦.

34 If x thousand dollars is spent on labor and y thousand dollars is spent on

equipment, the output at a certain factory will be Qxy( ,60)= x y13 2 3units. If

$120,000 is available, how should this be allocated between labor and equipment

to generate the largest possible output? Use Lagrange multiplier λto estimate the

change in the maximum output of the factory that will result if the money

available for labor and equipment is increased by $1,000. ebIeKcMNayxBan;duløa eTA

elIkMlaMgBlkmμ nigy Ban;duløaeTAelIsmÖar³ enaHTinñplecjBIeragcRkmYykMNt;edayGnuKmn_

()

Qxy,60= x y13 2 3Ékta. ebIman 120000 duløasRmab;cMNay enaHetIeKKYrcMNayelIkMlaMg

Blkmμb:unμan nigelIsmÖar³b:unμanedIm,I[)anTinñplx<s;bMput? eRbIemKuNLaRkg; λedIm,I):an;sμan

nUvTMhMERbRbYlrbs;kMritGtibrmaénTinñplrbs;eragcRkebIeKbegáIncMNaycMnYn 1000 duløa.

35 A consumer has $280 to spend on two commodities, the first of which costs $2
per unit and the second $5 per unit. Suppose that the utility derived by the
consumer from x units of the first commodity and y units of the second
commodity is. How many units of each commodity should

the consumer buy to maximize utility?GtifiCnmñak;)ancMNay 280 duløaeTAelIrbs; eRbI

R)as;BIrRbePT. RbePTTI1 éfø 2 duløakñúg1Ékta nigRbePTTI2éfø 5 duløakñúg1Ékta. ]bmafa

GtßRbeyaCn_Edl)anmkBIrbs;eRbIR)as;RbePTTI1cMnYnxÉkta nigBIrbs;eRbIR)as;RbePTTI2cMnYny

ÉktakMNt;edayGnuKmn_. etIGtifiCnKYrTijrbs;eRbIR)as;RbePT

nimYy²b:unμanÉktaeFVI[GnuKmn_GtßRbeyaCn_Edl)anmk mankMritGtibrma?

Uxy(), = 100 x y0.25 0.75

Uxy(), = 100 x y0.25 0.75

36 A consumer has k dollars to spend on two commodities, the first of which costs a
dollars per unit and the second b dollars per unit. Suppose that the utility derived
by the consumer from x units of the first commodity and y units of the second