Calculo 2 De dos variables_9na Edición - Ron Larson

13.9 Applications of Extrema of Functions of Two Variables 967
SECCIÓN 13.9 Aplicaciones de los extremos de funciones de dos variables 967
12. Maximum Volume Show that the rectangular box of maximum
volume máximo inscribed Mostrar in a sphere que la caja of radius rectangular r is a cube. de volumen 13.9 13.9Applications 19. point Costo S
19. Minimum Cost A water line is to be built from point P to
12. Volumen of mínimo and
Extrema of must Extrema Hay pass
of que Functions
through of construir Functions regions
of un Two of conducto where Two Variables
construction Variables para agua costs desde 967 967
máximo inscrita en una esfera de radio r es un cubo.
el punto P al punto S y debe atravesar regiones donde los costos
13. Volume and Surface Area Show that a rectangular box of differ (see figure). The cost per kilometer in dollars is 3k from
13. Volumen y área exterior Mostrar que una caja rectangular de
de construcción difieren (ver la figura). El costo por kilómetro
given volume and minimum surface area is a cube.
P to Q, 2k from Q to R, and k from R to S. Find x and y such
12.
volumen
12. Maximum Maximum Volume
dado y
Volume Show
área exterior
Show that
mínima
that the rectangular the
es un
rectangular box
cubo.
box of maxi-
of maxi-
volume A volume trough inscribed with inscribed trapezoidal a in sphere a sphere cross of radius of sections radius r is a ris cube. is formed a cube. by point tales point que S and el S must and costo must pass total pass through C se through minimice. regions regions where where construction costs costs
19. 19. Minimum that en dólares Minimum the total Cost es cost 3kCost de A CPwill water a A Q, be water 2k line minimized. de line is Qto a is Rbe to y built kbe de built Rfrom a S. from point Hallar point Px to y Py
to
14. Areamum
14. Área turning Un up comedero the edges of de a secciones 30-inch-wide transversales sheet of aluminum en forma (see de
13.
trapecio
13. Volume Volume and
se forma
and Surface
doblando
Surface Area Area Show
los extremos
Show that that a rectangular
de una
a rectangular box
lámina de aluminio
box of of
differ differ (see P (see figure). figure). The The cost cost per kilometer per kilometer in dollars in dollars is 3kis
from 3k from
figure). Find the cross section of maximum area.
given given volume
de 30
volume and
pulgadas
and minimum
de
minimum surface
ancho (ver
surface area
la figura).
area is a cube. is
Hallar
a cube.
P to PQ,
to 2k Q, from 2k from Q to QR,
to and R, and k from k from R to RS.
to Find S. Find x and x and y such y such
la sección that 2 kmthat the total the total cost cost C will C will be minimized. be minimized.
14. transversal 14. Area Area A trough de A área trough with máxima. with trapezoidal trapezoidal cross cross sections sections is formed is formed by by 2 km
x Q
turning x
xturning up the up edges the edges of x a of 30-inch-wide a sheet sheet of aluminum of aluminum (see (see 1 km
1 km
P P R
figure). figure). Find Find the cross the cross section section of maximum of maximum area. area.
y
θ x
x θ
y
S
2 km2 km
S
θ
30 − 2x
θ
x Q
x10 Qkm
10 km
x x
x x
1 km1 km R R
15. Maximum 30 Revenue − 2x A company manufactures two types of 20. Distance A company has retail outlets located at the points
20. Distanciay
Una y empresa tiene tres tiendas de ventas al menudeo
θ
θ
S
sneakers, θ running shoes θ and basketball shoes. The total revenue 0, 0 , 2, 2 , and 2, 2 S(see figure). Management plans to
15. Ingreso máximo Una empresa fabrica dos tipos de zapatos localizadas en los puntos (0, 0), (2, 2) y (2, 2) (ver la figura). La
from x
tenis, tenis 1 units 30 − 30 2x of − running 2x shoes and x
para correr y tenis para baloncesto. 2 units of basketball shoes build a distribution 10 km10 kmcenter located such that the sum of the
El ingreso total de dirección planea construir un centro de distribución localizado de
is
2 2
R 5x
x unidades de 1 8x
tenis 2 2x
para correr 1 x 2 42x
15. Maximum Revenue A company y x unidades 1 102x
de 2 , where x
tenis de baloncesto
2 are in thousands of units. Find x
1 and distances S from the center to the outlets is minimum. From the
15. Maximum tal manera que la suma S de las distancias del centro a las tiendas
x 1 Revenue A company 2 manufactures two two types types of of 20. 20. Distance Distance A company A company has retail has retail outlets outlets located located the at points the points
sneakers, es R 5x running 12 shoes 8x 22 and 2x 1 basketball x 1 and x
2 42x 2 so as to maximize symmetry of the problem it is clear that the distribution center
sneakers, 1 shoes. 102x The 2 , total donde revenue x sea mínimo. Por la simetría del problema es claro que el centro de
the revenue. running shoes and basketball shoes. The total revenue 1 will 0, 0 be 0, , 0 located 2, , 22, , and 2 on , and the 2, y-axis, 2 2, (see 2and (see figure). therefore figure). Management S is a function plans of plans the to to
y from x 2 están from x units en x miles de unidades. Hallar las y que maximizan distribución se localizará el eje y, y por consiguiente S es una
1 units of running of running shoes shoes and and x units x x
2
1 units of xbasketball 2 of basketball shoes shoes build build a distribution a distribution center center located located such such that that the sum the sum of the of the
16. el Maximum 1
ingreso. Profit
is
2
is
2 2A corporation
2
manufactures 2
candles at two
single variable y. Using techniques presented in Chapter 3, find
R R 5x where and
función de una variable y. Utilizando las técnicas presentadas en el
1 5x 1
8x 2 8x 2
2x 1 x2x 2 1 x 2
42x 1 42x 1
102x 102x 2 , where 2 , x and x 1
distances distances S from S from the center the center to the to outlets the outlets is minimum. is minimum. From From the the
locations. The cost of producing x units at location 1 is 1
the required value of y.
16. Ganancia x are x are o in beneficio thousands máximo of units. Find Una empresa and fabrica so as to velas maximize en
capítulo 3, calcular el valor de y requerido.
2
in thousands of units. Find 1
2 x 1 and x 1
x 2 so x 2
as to maximize symmetry symmetry of the of problem the problem it is it clear is clear that that the distribution the distribution center center
y
y
dos the lugares. the revenue. El costo de producción de unidades en el lugar 1
C revenue.
x 1
will will be located be located on the on the axis, y-axis, and therefore and therefore is a Sfunction is a function of the
1 0.02x
2
1 4x 1 500
y-
S
of the
16. es 16. Maximum Maximum Profit Profit A corporation A corporation manufactures candles candles at two
single
at two
single variable 3variable
y. Using y. Using techniques techniques presented 4
presented in Chapter in Chapter 3, find 3, find
and the cost of producing x units at location 2 is
(−2, 2)
(2, (2, 2) 2)
locations. The cost of producing units location 1 is
the required value of y.
C
locations. x 1 1 0.02x
The
12
cost
4x
of
1
producing 2
500
x units at location 1 is
the required 2 value d of y.
1
3
(−2,
(−2, 2)
2) (4,
(4,
2)
d
2)
(x,
(x,
y)
y)
C 2 y y
y y
2 0.05x
2
(0, y)
y el costo
C 1 de
0.02x
2 1 0.02x 2 4x
1
producción 1
4x 2 275.
1 4x 1
500
de
500
d
(0, y)
x 2
unidades en el lugar 2 es
1
d 2 d
1 x
d 3
3
2
−3
(0, 0)
3
4 4 1
The candles sell for $15 per unit. Find the quantity that should
x
and the cost of producing units location 2 is
(−2, 2)
(2, 2)
C
and x 2
2
the
0.05x
cost of
22
producing
4x 2 275.
units at location 2 is
(−2, −3
2) (0, 0) 1 (2, 2
x
2)
2
2 d2
d −2 (0, (0,
0)
0) 2 4
be produced at each location to maximize the profit
3 3
(−2, (−2, 2) 2) (4, 2) (4, 2)
−2d (x, y)
−2
Las
C 2
velas 2 se
0.05x
2 d −2
(x, y)
CP venden 4x
a 2 $15
275.
2 2
15 0.05x x 2
2 4x 2 275.
(0, y)
−2
1 x 2 C 1 C
por 2 .
unidad. Hallar la cantidad que
d (0, y)
1 d 1
d 2 d 2 d d
d d 3 3
17. debe Hardy-Weinberg
The producirse candles en Law
sell cada for $15 lugar Common
per para blood
unit. aumentar types
Find the al are
quantity máximo determined
The candles sell for $15 per unit. Find the quantity that that should el beneficio
genetically −2 (0, 0) 2 4
should Figure 1 1
Figura −3 −2 −3(0, for para −20)
(0, 20 20 0) 1 21
2 Figure Figura for para 21 21
be Pproduced by 15x three
1 alleles
at x 2 each CA, 1 location B, Cand 2 . O. (An allele is any of a
−2 (0, 0) 2 4
be produced at each location to maximize to maximize the the profit profit
group of possible mutational forms of a gene.) A person whose CAS 21. Investigación Investigation −2 −2 The Las tiendas retail outlets de ventas described al menudeo in Exercise descritas 20 en arel
17. Ley P de P15Hardy-Weinberg x15 x Los tipos sanguíneos son genéticamente
17. Hardy-Weinberg determinados por Law tres Common alelos blood A, B types y O. are (Alelo determined es
−2
1 x 2 C 1 C 2 .
−2
1 x 2 C 1 C 2 .
blood type is AA, BB, or OO is homozygous. A person whose ejercicio located at 200, se 0 localizan , 4, 2 , en and (0, 0), 2, (4, 2 2) (see y (2, figure). 2) (ver The la location figura).
17. blood type is AB, Law AO, Common or BO is blood heterozygous. types are The determined Hardy- Figure La of the localización del centro de distribución es (x, y), y por consiguiente
distances la Ssuma is a function S de las distancias of x and y. es una función de x y y.
cualquiera genetically de las by posibles three alleles formas A, de B, mutación and O. (An de un allele gen.) is any Una
Figure distribution for 20 for 20 center is x, y Figure , and Figure therefore 21
21 the sum of the
genetically Weinberg Law by three states alleles that A, the B, proportion and O. (An P allele of heterozygous is any of a of a
persona group group of cuyo possible of possible tipo mutational sanguíneo mutational es forms AA, forms of BB a of gene.) u OO a gene.) es A homocigótica.
person A person whose whose CAS CAS
individuals in any given population is
21. 21. Investigation The The retail retail outlets outlets described described in Exercise in Exercise 20 are 20 are
Una persona cuyo tipo sanguíneo es AB, AO o BO es heterocigótica.
La ley Hardy-Weinberg establece que la proporción P
(a) Escribir Write the la expression expresión giving que da the la suma of the S de distances las distancias. S. Use
blood blood type type is AA, is AA, BB, BB, or OO or OO is homozygous. is A person A person whose whose located located 0, at 00, , 04, , 24, , and 2 , and 2, 2 2, (see 2 (see figure). figure). The The location location
Utilizar un sistema algebraico por computadora y representar
blood Pp, blood q, type r type is 2pq AB, is AB, 2prAO, or 2qr
a computer algebra system to graph S. Does the surface
BO or BO is heterozygous. is The The Hardy- Hardy- of the of distribution the distribution center center is x, is y x, , and y , and therefore therefore sum the sum of the of the
de individuos heterocigótica en cualquier población dada es
S. have ¿Tiene a minimum? esta superficie un mínimo?
Weinberg Weinberg Law Law states states that that the proportion the proportion P of Pheterozygous
of distances distances S
Sa function is a function of x of and x and y. y.
where p represents the percent of allele A in the population, q
individuals in any given population is
Utilizar un sistema algebraico por computadora obtener S
(a) Write the expression giving the sum of the distances S. Use x
P
individuals
p, q, r 2pq
any
given
2pr
population
2qr
is
(b) Use a computer system to obtain S
(a) Write the expression giving the sum of the x and S
distances y . Observe
represents the percent of allele B in the population, and r
S. Use
y that S y . Observar solving the que system resolver S el sistema x y S y 0 es muy
a computer a computer algebra algebra system system x 0 and S
to graph to graph y 0 is very difficult.
Pp, represents S. Does the surface
donde
Pp, q, r
p representa
q, the r2pq percent 2pq2pr of
el porcentaje
2pr allele 2qrO 2qr in the population. Use the fact
S. Does the surface
de alelos A en la población, q
difícil. So, approximate Por tanto, aproximar the location la of localización the distribution del centro center. de distribución.
that p q r 1 to show that the maximum proportion of
have have a minimum? a minimum?
representa el porcentaje de alelos B en la población y r representa
represents el porcentaje the de percent alelos O of en allele la población. B in the Utilizar population, el hecho and c) Una estimación inicial del punto crítico es x x
where where p represents p represents the percent the percent of allele of allele A in A the 2in population, the population, q q (c) An initial estimate of the critical point is x
heterozygous individuals in any population is
(b) Use (b) Use a computer a computer algebra algebra system system to obtain to obtain S and 1 , y
S and S Observe
S 1 1, 1 .
3.
x Observe
1 , y .
1 y .
represents the percent of allele B in the population, and r r Calculate S 1, 1 with components S 1, 1.
that that solving solving the system the system S S and S is very difficult.
de que represents the percent para of allele mostrar O in que the population. proporción Use máxima
that de that individuos p q heterocigóticos r to 1show to show that en that the cualquier maximum the maximum población proportion proportion es 3 .
So, y 1, 1 . What direction is given by the vector S 1 ?
the fact Calcular S1, 1 con componentes x
and 0
S y
is very 0 difficult.
represents p the q percent r 1
x 0 S y 0 x 1, 1 and
18. Shannon Diversity Index One way to measure species diversity
is to use the Shannon diversity index H. If a habitat consists
approximate the location of the distribution center.
of allele O in the population. Use the fact
S x 1, 1 y S y 1, 1.
2
So, the location of the distribution center.
p q r 1
of of (d) ¿Qué The second dirección estimate la dada of the por critical el vector point S1, is 1?
of three species, A, B, and C, its Shannon diversity 2 2 index is
(c) An (c) initial An initial estimate estimate of the of critical the critical point point is xis
x
18. Índice heterozygous de diversidad individuals individuals de Shannon any in population any Una population forma is de is medir diversidad
H
d) La segunda estimación del punto crítico es 1 , y 11 , y 1
1, 11, . 1 .
3. 3.
Calculate Calculate S 1, S1
1, with with components S S and
18. de Shannon especies x ln x
Diversity es y usar ln y el Index índice z ln z
x
de One diversidad way to measure de Shannon species H. Si
2 , y 2 x 1 S x x 1 , y 1 t, y 1 S y x 1 , y 1 t . x 1, 1 x 1, and 1
18. Shannon Diversity Index One way to measure species diversity
where is sity to xis use to the use Shannon percent the Shannon of diversity diversity index A index
H. the If H. a habitat, If a habitat consists y
consists the
If these coordinates are substituted into S x, y , then S
diver-
x
un hábitat consiste de tres especies, A, B y C, su índice de diversidad
of percent three of de three species, Shannon of species, A, es
(d)
2
S, y y1, 2 S 1 y 1, . x What 1 . What direction S x x 1
direction , y 1 t, is ygiven 1
is given Sby y xthe 1 , by y 1
vector the t. vector S 1, S1 1, ? 1 ?
B, B A, and in B, the and C, its habitat, C, Shannon its Shannon and diversity z is diversity the index percent index is of is
Si The (d) se The second sustituyen second estimate estimate estas of coordenadas the of critical the critical point en Sx, point isy,
is
becomes a function of the single variable t. Find entonces the value S se
species C in the habitat.
convierte en una función de una variable t. Hallar el valor de
H H x ln x ln xy ln y ln yz ln z
ln z
of x 2 , t ythat x 2
t que minimiza 2 , yminimizes
2
x 1 x
S. 1
S x
S. x 1 S, Use
Utilizar x
yx 11 t, , this y
este 1 t, value y
valor 1
S y xof 1 S, de t y
tyto x 1
para 1
t , estimate y.
estimar 1 t . x 2 , y 2 .
(a) Use the fact that x y z 1 to show that the maximum
x 2 , y 2 .
donde where where xes is el xthe porcentaje is percent the percent de of especies of species A en in A el the in hábitat, habitat, the habitat, y es y el is ypor-
centaje percent percent de of especies of H 1
the is the (e) Complete two more iterations of the process in part (d) to
value of occurs when
e) Realizar If these If these dos coordinates iteraciones coordinates are más are substituted del substituted proceso into del into S x, inciso Sy x, , then y d) , para then S S
x y z
species B in en B the in el habitat, the hábitat habitat, y and 3
z and es z
. is el zthe porcentaje is percent the percent of de of obtain x
obtener becomes becomes x 4 , y
a 4 , function y 4 . For this location of the distribution center,
4
a . function Dada of esta the of single the localización single variable variable del t. Find centro t. Find the de value the distribución,
of t that of t that minimizes ¿cuál minimizes
la S. suma Use S. Use this de value this las distancias value of t to of estimate t to a estimate las tiendas x 2 , yx 22 . al , y 2 .
value
especies (b) species Use C the in en C results the el in hábitat. habitat. the of habitat. part (a) to show that the maximum value
what is the sum of the distances to the retail outlets?
of H in this habitat is ln 3.
(a) Usar Use (a) Use el the factor the fact that de that x + xy+ z yz= 1 z1
para to 1show demostrar to show that that the que maximum the el maximum valor
(f)
(e) menudeo?
Explain why Sx, y was used to approximate the
Complete (e) Complete two two more more iterations iterations of the of process the process in part in part (d) to (d) to
1
máximo value value of de H of Hoccurs H 1
ocurre occurs when cuando when x xy yz
z3 . 3 .
minimum value of S. In what types of problems would you
f) Explicar obtain obtain xpor 4 , yx 4 qué 4
., yFor 4 se . For this usó this location Sx, location y of para the of distribution the aproximar distribution center, el valor center,
use S x, y ?
(b) Usar Use (b) el Use the resultado results the results of del part of inciso part (a) to (a) show to para show that demostrar that the maximum the que maximum el value valor value mínimo what what is de the is S. sum the ¿En sum of qué the of tipo distances the de distances problemas to the to retail the se usaría retail outlets? outlets? Sx, y?
máximo of Hof
in Hde this in Hthis habitat en este habitat is hábitat ln is 3. ln es 3. de ln 3.
(f) Explain (f) Explain why why Sx, Sx, y was y was used used to approximate to the the
minimum minimum value value of S. of In S. what In what types types of problems of problems would would you you
use use S x, Sy x, ? y ?