Calculus for Business, Economics, Life Sciences and Social Sciences

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Chapter 8

Multivariable Calculus - all with Video Answers

Educators

Section 1

Functions of Several Variables

Problem 1

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
f(4,-1)
$$

Dwijendra Rao

Dwijendra Rao

Numerade Educator

Problem 2

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
f(0,10)
$$

Dwijendra Rao

Numerade Educator

Problem 3

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
f(8,0)
$$

Dwijendra Rao

Numerade Educator

Problem 4

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
f(5,6)
$$

Dwijendra Rao

Numerade Educator

Problem 5

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
g(1,7)
$$

Dwijendra Rao

Numerade Educator

Problem 6

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
g(-2,0)
$$

Dwijendra Rao

Numerade Educator

Problem 7

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
g(3,-3)
$$

Dwijendra Rao

Numerade Educator

Problem 8

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
g(0,0)
$$

Dwijendra Rao

Numerade Educator

Problem 9

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
3 f(-2,2)+5 g(-2,2)
$$

Lucas Finney

Numerade Educator

Problem 10

Find the indicated values of the functions $f(x, y)=2 x+7 y-5 \quad$ and $\quad g(x, y)=\frac{88}{x^{2}+3 y}$.
$$
2 f(10,-4)-7 g(10,-4)
$$

Lucas Finney

Numerade Educator

Problem 11

Find the indicated values of $f(x, y, z)=2 x-3 y^{2}+5 z^{3}-1$.
$$
f(0,0,0)
$$

Dwijendra Rao

Numerade Educator

Problem 12

Find the indicated values of $f(x, y, z)=2 x-3 y^{2}+5 z^{3}-1$.
$$
f(0,0,2)
$$

Dwijendra Rao

Numerade Educator

Problem 13

Find the indicated values of $f(x, y, z)=2 x-3 y^{2}+5 z^{3}-1$.
$$
f(6,-5,0)
$$

Dwijendra Rao

Numerade Educator

Problem 14

Find the indicated values of $f(x, y, z)=2 x-3 y^{2}+5 z^{3}-1$.
$$
f(-10,4,-3)
$$

Dwijendra Rao

Numerade Educator

Problem 15

Find the indicated value of the given function.
$P(13,5)$ for $P(n, r)=\frac{n !}{(n-r) !}$

Dwijendra Rao

Numerade Educator

Problem 16

Find the indicated value of the given function.
$C(13,5)$ for $C(n, r)=\frac{n !}{r !(n-r) !}$

Dwijendra Rao

Numerade Educator

Problem 17

Find the indicated value of the given function.
$V(4,12)$ for $V(R, h)=\pi R^{2} h$

Dwijendra Rao

Numerade Educator

Problem 18

Find the indicated value of the given function.
$T(4,12)$ for $T(R, h)=2 \pi R(R+h)$

Dwijendra Rao

Numerade Educator

Problem 19

Find the indicated value of the given function.
$S(3,10)$ for $S(R, h)=\pi R \sqrt{R^{2}+h^{2}}$

Dwijendra Rao

Numerade Educator

Problem 20

Find the indicated value of the given function.
$W(3,10)$ for $W(R, h)=\frac{1}{3} \pi R^{2} h$

Dwijendra Rao

Numerade Educator

Problem 21

Find the indicated value of the given function.
$A(100,0.06,3)$ for $A(P, r, t)=P+\operatorname{Prt}$

Lucas Finney

Numerade Educator

Problem 22

Find the indicated value of the given function.
$A(10,0.04,3,2)$ for $A(P, r, t, n)=P\left(1+\frac{r}{n}\right)^{t n}$

Dwijendra Rao

Numerade Educator

Problem 23

Find the indicated value of the given function.
$P(0.05,12)$ for $P(r, T)=\int_{0}^{T} 4,000 e^{-r t} d t$

Dwijendra Rao

Numerade Educator

Problem 24

Find the indicated value of the given function.
$F(0.07,10)$ for $F(r, T)=\int_{0}^{T} 4,000 e^{r(T-t)} d t$

Dwijendra Rao

Numerade Educator

Problem 25

Find the indicated function fof a single variable.
$$
f(x)=G(x, 0) \text { for } G(x, y)=x^{2}+3 x y+y^{2}-7
$$

Lucas Finney

Numerade Educator

Problem 26

Find the indicated function fof a single variable.
$$
f(y)=H(0, y) \text { for } H(x, y)=x^{2}-5 x y-y^{2}+2
$$

Lucas Finney

Numerade Educator

Problem 27

Find the indicated function fof a single variable.
$$
f(y)=K(4, y) \text { for } K(x, y)=10 x y+3 x-2 y+8
$$

Lucas Finney

Numerade Educator

Problem 28

Find the indicated function fof a single variable.
$$
f(x)=L(x,-2) \text { for } L(x, y)=25-x+5 y-6 x y
$$

Lucas Finney

Numerade Educator

Problem 29

Find the indicated function fof a single variable.
$$
f(y)=M(y, y) \text { for } M(x, y)=x^{2} y-3 x y^{2}+5
$$

Lucas Finney

Numerade Educator

Problem 30

Find the indicated function fof a single variable.
$$
f(x)=N(x, 2 x) \text { for } N(x, y)=3 x y+x^{2}-y^{2}+1
$$

Lucas Finney

Numerade Educator

Problem 31

Let $F(x, y)=2 x+3 y-6 .$ Find all values of $y$ such that $F(0, y)=0$.

Dwijendra Rao

Numerade Educator

Problem 32

Let $F(x, y)=5 x-4 y+12$. Find all values of $x$ such that $F(x, 0)=0$.

Dwijendra Rao

Numerade Educator

Problem 33

Let $F(x, y)=2 x y+3 x-4 y-1$. Find all values of $x$ such that $F(x, x)=0$.

Dwijendra Rao

Numerade Educator

Problem 34

Let $F(x, y)=x y+2 x^{2}+y^{2}-25 .$ Find all values of $y$ such that $F(y, y)=0$.

Dwijendra Rao

Numerade Educator

Problem 35

Let $F(x, y)=x^{2}+e^{x} y-y^{2} .$ Find all values of $x$ such that $F(x, 2)=0$.

Dwijendra Rao

Numerade Educator

Problem 36

Let $G(a, b, c)=a^{3}+b^{3}+c^{3}-(a b+a c+b c)-6 .$ Find all values of $b$ such that $G(2, b, 1)=0$.

Dwijendra Rao

Numerade Educator

Problem 37

For the function $f(x, y)=x^{2}+2 y^{2},$ find $$\frac{f(x+h, y)-f(x, y)}{h}$$.

Dwijendra Rao

Numerade Educator

Problem 38

For the function $f(x, y)=x^{2}+2 y^{2},$ find $\frac{f(x, y+k)-f(x, y)}{k}$

Dwijendra Rao

Numerade Educator

Problem 39

For the function $f(x, y)=2 x y^{2},$ find $$\frac{f(x+h, y)-f(x, y)}{h}$$

Dwijendra Rao

Numerade Educator

Problem 40

For the function $f(x, y)=2 x y^{2},$ find $$\frac{f(x, y+k)-f(x, y)}{k}$$.

Dwijendra Rao

Numerade Educator

Problem 41

Find the coordinates of $E$ and $F$ in the figure for Matched Problem 6 on page $453 .$

Dwijendra Rao

Numerade Educator

Problem 42

Find the coordinates of $B$ and $H$ in the figure for Matched Problem 6 on page $453 .$

Dwijendra Rao

Numerade Educator

Problem 43

Use a graphing calculator as necessary to explore the graphs of the indicated cross sections.
Let $f(x, y)=x^{2}$
(A) Explain why the cross sections of the surface $z=f(x, y)$ produced by cutting it with planes parallel to $y=0$ are parabolas.
(B) Describe the cross sections of the surface in the planes $x=0, x=1,$ and $x=2$
(C) Describe the surface $z=f(x, y)$.

Sunanda Adibhatla

Sunanda Adibhatla

Numerade Educator

Problem 44

Use a graphing calculator as necessary to explore the graphs of the indicated cross sections.
Let $f(x, y)=\sqrt{4-y^{2}}$
(A) Explain why the cross sections of the surface $z=f(x, y)$ produced by cutting it with planes parallel to $x=0$ are semicircles of radius 2 .
(B) Describe the cross sections of the surface in the planes $y=0, y=2,$ and $y=3$
(C) Describe the surface $z=f(x, y)$.

Sunanda Adibhatla

Sunanda Adibhatla

Numerade Educator

Problem 45

Use a graphing calculator as necessary to explore the graphs of the indicated cross sections.
Let $f(x, y)=\sqrt{36-x^{2}-y^{2}}$
(A) Describe the cross sections of the surface $z=f(x, y)$ produced by cutting it with the planes $y=1, y=2$, $y=3, y=4,$ and $y=5$
(B) Describe the cross sections of the surface in the planes $x=0, x=1, x=2, x=3, x=4,$ and $x=5$
(C) Describe the surface $z=f(x, y)$.

Sunanda Adibhatla

Sunanda Adibhatla

Numerade Educator

Problem 46

Use a graphing calculator as necessary to explore the graphs of the indicated cross sections.
Let $f(x, y)=100+10 x+25 y-x^{2}-5 y^{2}$
(A) Describe the cross sections of the surface $z=f(x, y)$ produced by cutting it with the planes $y=0, y=1$, $y=2,$ and $y=3$
(B) Describe the cross sections of the surface in the planes $x=0, x=1, x=2,$ and $x=3$
(C) Describe the surface $z=f(x, y)$.

Sunanda Adibhatla

Sunanda Adibhatla

Numerade Educator

Problem 47

Use a graphing calculator as necessary to explore the graphs of the indicated cross sections.
Let $f(x, y)=e^{-\left(x^{2}+y^{2}\right)}$
(A) Explain why $f(a, b)=f(c, d)$ whenever $(a, b)$ and $(c, d)$ are points on the same circle centered at the origin in the $x y$ plane.
(B) Describe the cross sections of the surface $z=f(x, y)$ produced by cutting it with the planes $x=0, y=0$, and $x=y$
(C) Describe the surface $z=f(x, y)$

Sunanda Adibhatla

Sunanda Adibhatla

Numerade Educator

Problem 48

Use a graphing calculator as necessary to explore the graphs of the indicated cross sections.
Let $f(x, y)=4-\sqrt{x^{2}+y^{2}}$
(A) Explain why $f(a, b)=f(c, d)$ whenever $(a, b)$ and $(c, d)$ are points on the same circle with center at the origin in the $x y$ plane.
(B) Describe the cross sections of the surface $z=f(x, y)$ produced by cutting it with the planes $x=0, y=0$, and $x=y$
(C) Describe the surface $z=f(x, y)$.

Sunanda Adibhatla

Sunanda Adibhatla

Numerade Educator

Problem 49

A small manufacturing company produces two models of a surfboard: a standard model and a competition model. If the standard model is produced at a variable cost of $\$ 210$ each and the competition model at a variable cost of $\$ 300$ each, and if the total fixed costs per month are $\$ 6,000$, then the monthly cost function is given by
$$
C(x, y)=6,000+210 x+300 y
$$
where $x$ and $y$ are the numbers of standard and competition models produced per month, respectively. Find $C(20,10)$, $C(50,5),$ and $C(30,30)$

Dwijendra Rao

Numerade Educator

Problem 50

A company spends $\$ x$ thousand per week on online advertising and $\$ y$ thousand per week on TV advertising. Its weekly sales are found to be given by
$$
S(x, y)=5 x^{2} y^{3}
$$
Find $S(3,2)$ and $S(2,3)$.

Dwijendra Rao

Numerade Educator

Problem 51

A supermarket sells two brands of coffee: brand $A$ at $\$ p$ per pound and brand $B$ at $\$ q$ per pound. The daily demand equations for brands $A$ and $B$ are, respectively,
$$
\begin{array}{l}
x=200-5 p+4 q \\
y=300+2 p-4 q
\end{array}
$$
(both in pounds). Find the daily revenue function $R(p, q)$ Evaluate $R(2,3)$ and $R(3,2)$

Dwijendra Rao

Numerade Educator

Problem 52

A company manufactures 10 - and 3 -speed bicycles. The weekly demand and cost equations are
$$
\begin{aligned}
p &=230-9 x+y \\
q &=130+x-4 y \\
C(x, y) &=200+80 x+30 y
\end{aligned}
$$
where $\$ p$ is the price of a 10 -speed bicycle, $\$ q$ is the price of a 3-speed bicycle, $x$ is the weekly demand for 10 -speed bicycles, $y$ is the wee kly demand for 3 -speed bicycles, and $C(x, y)$ is the cost function. Find the weekly revenue function $R(x, y)$ and the weekly profit function $P(x, y) .$ Evaluate $R(10,15)$ and $P(10,15)$.

Lucas Finney

Numerade Educator

Problem 53

The Cobb-Douglas production function for a petroleum company is given by
$$
f(x, y)=20 x^{0.4} y^{0.6}
$$
where $x$ is the utilization of labor and $y$ is the utilization of capital. If the company uses 1,250 units of labor and 1,700 units of capital, how many units of petroleum will be produced?

Lucas Finney

Numerade Educator

Problem 54

The petroleum company in Problem 53 is taken over by another company that decides to double both the units of labor and the units of capital utilized in the production of petroleum. Use the Cobb-Douglas production function given in Problem 53 to find the amount of petroleum that will be produced by this increased utilization of labor and capital. What is the effect on productivity of doubling both the units of labor and the units of capital?

Lucas Finney

Numerade Educator

Problem 55

At the end of each year, $\$ 5,000$ is invested into an IRA earning $3 \%$ compounded annually.
(A) How much will be in the account at the end of 30 years? Use the annuity formula
$$
F(P, i, n)=P \frac{(1+i)^{n}-1}{i}
$$
where
$$
\begin{aligned}
P &=\text { periodic payment } \\
i &=\text { rate per period } \\
n &=\text { number of payments (periods) } \\
F &=\mathrm{FV}=\text { future value }
\end{aligned}
$$
(B) Use graphical approximation methods to determine the rate of interest that would produce $\$ 300,000$ in the account at the end of 30 years.

Lucas Finney

Numerade Educator

Problem 56

The packaging department in a company has been asked to design a rectangular box with no top and a partition down the middle (see the figure on the next page). Let $x, y,$ and $z$ be the dimensions of the box (in inches).
(A) Find the total amount of material $M(x, y, z)$ used in constructing one of these boxes, and evaluate $M(10,12,6)$.
(B) Suppose that the box will have a square base and a volume of 720 cubic inches. Use graphical approximation methods to determine the dimensions that require the least amount of material.

Lucas Finney

Numerade Educator

Problem 57

Using scuba-diving gear, a marine biologist estimates the time of a dive according to the equation
$$
T(V, x)=\frac{33 V}{x+33}
$$
where
$T=$ time of dive in minutes $V=$ volume of air, at sea level pressure, compressed into tanks
$x=$ depth of dive in feet
Find $T(70,47)$ and $T(60,27)$

Lucas Finney

Numerade Educator

Problem 58

Poiseuille's law states that the resistance $R$ for blood flowing in a blood vessel varies directly as the length $L$ of the vessel and inversely as the fourth power of its radius $r$. Stated as an equation,
$$
R(L, r)=k \frac{L}{r^{4}} \quad k \text { a constant }
$$
Find $R(8,1)$ and $R(4,0.2)$.

Lucas Finney

Numerade Educator

Problem 59

Anthropologists use an index called the cephalic index. The cephalic index $C$ varies directly as the width $W$ of the head and inversely as the length $L$ of the head (both viewed from the top). In terms of an equation,
$$
C(W, L)=100 \frac{W}{L}
$$
where
$$
\begin{aligned}
W &=\text { width in inches } \\
L &=\text { length in inches }
\end{aligned}
$$
Find $C(6,8)$ and $C(8.1,9)$.

Lucas Finney

Numerade Educator

Problem 60

Under ideal conditions, if a person driving a car slams on the brakes and skids to a stop, the length of the skid marks (in feet) is given by the formula
$$
L(w, v)=k w v^{2}
$$
where
$$
\begin{aligned}
k &=\text { constant } \\
w &=\text { weight of car in pounds } \\
v &=\text { speed of car in miles per hour }
\end{aligned}
$$
For $k=0.0000133,$ find $L(2,000,40)$ and $L(3,000,60)$

Dwijendra Rao

Numerade Educator

Problem 61

The intelligence quotient (IQ) is defined to be the ratio of mental age (MA), as determined by certain tests, to chronological age (CA), multiplied by 100. Stated as an equation,
$$
Q(M, C)=\frac{M}{C} \cdot 100
$$
where
$$
Q=\text { IQ } \quad M=\mathrm{MA} \quad C=\mathrm{CA}
$$
Find $Q(12,10)$ and $Q(10,12)$.

Lucas Finney

Numerade Educator