Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

20 60+−x 70 y

40 8 5

Gav. etIm©as;hagKYrkMNt;éfølk;énGavRbePTnimYy²b:unμan edIm,I[)anR)ak;

cMeNjx<s;bMput?

22 The telephone company is planning to introduce two new types of executive
communications systems that it hopes to sell to its largest commercial customers.
It is estimated that if the first type of system is priced at x hundred dollars per
system and the second type at y hundred dollars per system, approximately
−+x yconsumers will buy the first type and 50 + 9 x− 7 ywill buy the
second type. If the cost of manufacturing the first type is $1,000 per system and
the cost of manufacturing the second type is $3,000 per system, how should the
telephone company price the systems to generate the largest possible profit?

Rkumh‘unTUrs½BÞmYyeRKagnwgdak;bgðajRbB½n§RbtibtþikarTUrKmnaKmn_fμIcMnYnBIrRbePTEdlBYkeKKit

faGacnwglk;eTA[GtifiCnBaNiC¢kmμFM²bMputrbs;BYkeK. eKrMBwgfa ebIsinCaRbB½n§RbePTTI1

lk;kñúgtémø x ryduløakñúg1RbB½n§ nigRbB½n§RbePTTI2lk;kñúgtémøyryduløakñúg1RbB½n§

enaHGtifiCncMnYnRbEhlCa40 8−+x 5 y nwgTijRbB½n§RbePT1 ehIyGtifiCncMnYnRbEhlCa

50 9+−x 7 ynak; nwgTijRbB½n§RbePTTI2. RbsinebIcMNayéføedImelIkarplitRbB½n§RbePTTI1

esIμnwg 1000 duløakñúgmYyRbB½n§ nigRbePTTI2esμInwg 3000 duløakñúgmYyRbB½n§ enaHetIRkumh‘unTUrs½BÞ

KYrkMNt;éfølk;b:unμanedIm,I[)anR)ak;cMeNjx<s;bMput?

23 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 the development and y thousand

dollars is spent on promotion, consumers will buy approimately

320 160

24

yx

yx

+

++

units of the product. If manufacturing costs for this product are $50 per unit, how

much should the manufacturer spend on development and how mnuch on

promotion to generate the largest possible profit from the sale of this product?

(Hint: Profit=(number of units)(price per unit−cost per unit)−totla amount spent

on development and promotion)plitkrmñak;eRKaglk;plitplfμImYyrbs;xøÜnkñúgtémø 50

duløakñúg1Ékta ehIyKat;rMBwgfaebIcMNayeTAelIkarplitcMnYnxBan;duløa nigelIkarpSBVpSaycMnYn

yBan;duløa enaHGtifiCnnwgTijcMnYnRbEhl

320 160

2

y

yx

+

4

x

+ +

Éktaénplitpl. ebIsinCacMNay

éføedImeTAelIplitplTaMgenHesμInwg 50 duløakñúg1Ékta etIplitkrKYrcMNayelIkarplitb:unμan

nigelIkarpSBVpSayb:unμan edIm,I[R)ak;cMeNjBIkarlk;plitplenHx<s;bMput?

(Answer: $4000 on development and $6000 on promotion)

24 A dairy produces whole milk and skim milk in quantities x and y gallons,
respectively. Suppose that the price of whole milk is p(xx)= 100 − and that of
skim milk isqy( )=− 100 yand assume that Cxy( , )=x xy y^2 ++^2 is the joint-
cost function of the commodities. What should x and y be in order to maximize
profit? kEnøgpliteRKOgbriePaKBITwkedaHeKamYyplitTwkedaHeKa whole milk nigTwkedaHeKa