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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
Demographers estimate that the fraction of people who will still be residing in a particular town t years after they arrive is given by the function f(t) = e−0.02t. If the current population is 50,000 and new townspeople arrive at the rate of 700 per year, what will the population be 20 years
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region bounded by the curve y = x2 − 2x and the x axis.
In Exercises 59 through 62, solve the given initial value problem.dx/dt = e−2t, where x = 4 when t = 0
Records indicate that t hours past midnight, the temperature at the local airport was f(t) = −0.3t2 + 4t + 10 degrees Celsius. What was the average temperature at the airport between 9:00 A.M. and noon?
In Exercises 37 through 44, sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed. L Vy+1 -1J-Vy+1 f(x, y) dy dx
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. S [1₂² 10 1-2 (2x + 3y) dy dx
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.In a learning experiment, a subject is first given x minutes to examine a list of facts. The fact sheet is then taken away and the subject is allowed y minutes to prepare mentally for an exam based on the fact
In Exercises 41 through 46, the demand functions for a pair of commodities are given. Use partial derivatives to determine whether the commodities are substitute, complementary, or neither. D₁ = 2,000 + D₂² 1,500 - 100 Pi + 2 P2 Pi + 7 25p2;
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. Lo fex exy dy dx -x-y
In Exercises 45 through 54, use a double integral to find the area of R.R is the triangle with vertices (−4, 0), (2, 0), and (2, 6).
Suppose a person of age A years has weight w in kilograms (kg) and height h in centimeters (cm). Then, the Harris-Benedict equations say that the daily basal energy expenditure in kilocalories is Bm(w, h, A) = 66.47 + 13.75w + 5.00h − 6.77A for a male and Bf(w, h, A) = 655.10 + 9.60w + 1.85h −
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Define the livable space of a building to be the volume of space in the building where a person 6 feet tall can walk upright. An A-frame cabin is y feet long and has equilateral triangular ends x feet on a
Output in a certain factory is given by the Cobb-Douglas production functionwhere K is the capital in $1,000 and L is the size of the labor force, measured in worker-hours.a. Use your calculator to obtain Q(K, L) for the values of K and L in this table:b. Note that the output Q(277, 743) is doubled
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.The beautiful patterns on the wings of butterflies have long been a subject of curiosity and scientific study. Mathematical models used to study these patterns often focus on determining the level of morphogen
Pediatricians and medical researchers sometimes use the following empirical formula* relating the surface area S (m2) of a person to the person’s weight W (kg) and height H (cm):a. Find S(15.83, 87.11). Sketch the level curve of S(W, H) that passes through (15.83, 87.11). Sketch several
The thin lens formula in optics says that the focal length L of a thin lens is related to the object distance do and image distance di by the equationIf L remains constant while do and di are allowed to vary, what is the maximum distance s = do + di between the object and the image? 1 1 + do d; 1 L
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. S. SAVT 0 xV1 - y dx dy
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Certain malignant tumors that do not respond to conventional methods of treatment such as surgery or chemotherapy may be treated by hyperthermia, which involves applying extreme heat to tumors using microwave
In Exercises 41 through 46, the demand functions for a pair of commodities are given. Use partial derivatives to determine whether the commodities are substitute, complementary, or neither. D₁ = 200p₁¹/2p₂-1/²; D₂ = 300p1-1/2p₂-3/2
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Tom, Dick, and Mary are participating in a cross-country relay race. Tom will trudge as fast as he can through thick woods to the edge of a river. Then Dick will take over and row to the opposite shore, where
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. [Lx -1 xe²y dy dx
A jewelry box is constructed by partitioning a box with a square base as shown in the accompanying figure. If the box is designed to have volume 800 cm3, what dimensions should it have to minimize its total surface area (top, bottom, sides, and interior partitions)? Notice that we have said nothing
Suppose the jewelry box in Exercise 48 is designed so that the material in the top costs twice as much as the material in the bottom and sides and three times as much as the material in the interior partitions. What dimensions minimize the total cost of constructing the box?Data from Exercises 48A
Having escaped from Blabba’s village before being detected the spy sneaks into Scelerat’s château. He enters a room and the door slams shut behind him. He begins to feel warm and realizes he is trapped inside Scelerat’s dreaded broiler room. Searching desperately for a way to survive, he
In Exercises 45 through 54, use a double integral to find the area of R.R is the triangle with vertices (0, −1), (−2, 1), and (2, 1).
A bicycle dealer has found that if 10-speed bicycles are sold for x dollars each and the price of gasoline is y cents per gallon, approximately F(x, y) bicycles will be sold each month, whereCurrently, the bicycles sell for $324 apiece and gasoline sells for $3.80 per gallon. Use marginal analysis
A particle of mass m in a rectangular box with dimensions x, y, and z has ground state energywhere k is a physical constant. In Exercise 52 of Section 7.3, you were asked to minimize the ground state energy subject to the fixed volume constraint V0 = xyz using substitution. Solve the same
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. блу² 1 X2 + 1 ITA -1 dy dx
A manufacturer estimates that the annual output at a certain factory is given byunits, where K is the capital expenditure in units of $1,000 and L is the size of the labor force in worker-hours.a. Find the marginal productivity of capital and the marginal productivity of labor when the capital
Dumping and other material handling operations near a landfill may result in contaminated particles being emitted into the surrounding air. To estimate such particulate emission, the following empirical formula* can be used:where E is the emission factor (pounds of particles released into the air
One of Poiseuille’s laws† says that the speed of blood V (cm/sec) flowing at a distance r (cm) from the axis of a blood vessel of radius R (cm) and length L (cm) is given bywhere P (dynes/cm2) is the pressure in the vessel. Suppose a particular vessel has radius 0.0075 cm and is 1.675 cm
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. "e "e In (xy) dy dx
At a certain factory, the daily output is Q(K, L) = 60K1/2L1/3 units, where K denotes the capital investment measured in units of $1,000 and L the size of the labor force measured in worker-hours. Suppose that the current capital investment is $900,000 and that 1,000 worker-hours of labor are used
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = 1/2x2 and y = 2x.
A machine based on wind energy generally converts the kinetic energy of moving air to mechanical energy by means of a device such as a rotating shaft, as in a windmill. Suppose we have wind of velocity v traveling through a wind-collecting machine with cross-sectional area A. Then, in physics* it
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = √x and y = x2.
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. (1-x SS x( y − 1)² dy dx 10 0
A rectangular building is to be constructed of material that costs $31 per square foot for the roof, $27 per square foot for the sides and the back, and $55 per square foot for the facing and glass used in constructing the front. If the building is to have a volume of 16,000 ft3, what dimensions
A grocer’s daily profit from the sale of two brands of cat food iscents, where x is the price per can of the first brand and y is the price per can of the second. Currently the first brand sells for 50 cents per can and the second for 52 cents per can. Use marginal analysis to estimate the change
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = x2 − 4x + 3 and the x axis.
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = x2 + 6x + 5 and the x axis.
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. fife 0 ex dy dx
In Exercises 50 through 52, assume that the required extreme value is a relative extremum.A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is to be 6,400 yd2, and no fencing is needed along the river bank. Find the dimensions of the pasture that
A person’s intelligence quotient (IQ) is measured by the functionwhere a is the person’s actual age and m is his or her mental age.a. Find I(12, 11) and I(16, 17).b. Sketch the graphs of several level curves of I(m, a). How would you describe these curves? I(m, a) = 100m a
In Exercises 50 through 52, assume that the required extreme value is a relative extremum.Suppose you wish to construct a rectangular box with a volume of 32 ft3. Three different materials will be used in the construction. The material for the sides costs $1 per square foot, the material for the
In Exercises 50 through 52, assume that the required extreme value is a relative extremum.A particle of mass m in a rectangular box with dimensions x, y, and z has ground state energywhere k is a physical constant. If the volume of the box satisfies xyz = V0 for constant V0, find the values of x,
In Exercises 52 and 53, evaluate the given double integral for the specified region R.where R is the rectangle with vertices (−1, 0), (2, 0), (2, 3), and (−1, 3). 6x²y dA, JR
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = ln x, y = 0, and x = e.
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = x, y = ln x, y = 0, and y = 1.
In Exercises 73 through 76, evaluate the double integral over the specified region R. Choose the order of integration carefully. R dA; R: Vy≤ x ≤ 1,0 ≤ y ≤ 1
In an electric circuit with two resistors of resistance R1 and R2 connected in parallel, the total resistance R is given by the formulaShow that 1 || -| R R₁ + 1 R₂
A bicycle dealer has found that if 10-speed bicycles are sold for x dollars apiece and the price of gasoline is y cents per gallon, then approximatelybicycles will be sold each month. If the price of bicycles varies between $289 and $324 during a typical month, and the price of gasoline varies
The ideal gas law says that for n moles of an ideal gas, PV = nRT, where P is the pressure exerted by the gas, V is the volume of the gas, T is the temperature of the gas, and R is a constant (the gas constant). Compute the product Ꮩ ᎧᎢ ᎧᏢ ᎧᎢ ᎧᏢ Ꮩ
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.Use the formula obtained in Exercise 84 to find the slope of the level curve x2 + xy + y3 = 1 at the point (−1, 1). What is the equation
A manufacturer estimates that when x units of a particular commodity are sold domestically and y units are sold to foreign markets, the profit is given byhundred dollars. If monthly domestic sales vary between 100 and 125 units and foreign sales between 70 and 89 units, what is the average monthly
For the soft drink can in Exercise 77, the surface area is given by S = 2πR2 + 2πRH. Use calculus to estimate the change in surface that results if:a. The radius is increased from 3 to 4 cm while the height stays at 12 cm.b. The height is decreased from 12 to 11 cm while the radius stays at 3 cm.
A consultant determines that the density of people positively influenced by an advertising campaign in the region shown in the accompanying figure isp(x, y) = x ln y thousand people per square mile at the grid point (x, y) with x and y in miles. How many people in the test region are positively
A soft drink can is a cylinder H cm tall with radius R cm. Its volume is given by the formula V = πR2H. A particular can is 12 cm tall with radius 3 cm. Use calculus to estimate the change in volume that results if the radius is increased by 1 cm while the height remains at 12 cm.
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.Suppose y = h(x) is a differentiable function of x and that F(x, y) = C for some constant C. Use the chain rule (with x taking the role of
At a certain factory, output Q is related to inputs x and y by the expression Q(x, y) = 2x3 + 3x2y + y3 If 0 ≤ x ≤ 5 and 0 ≤ y ≤ 7, what is the average output of the factory?
Sandy Gibbons is an investment advisor with a new product she has been actively promoting. She estimates that in the region shown in the accompanying region, the density of interest is p(x, y) = y2e−0.5x people per square mile at the grid point (x, y) with x and y in miles. How many people in the
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.A soft drink can is H centimeters (cm) tall and has a radius of R cm. The cost of material in the can is 0.0005 cents per cm2, and the soda
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.A cylindrical silo set on a concrete block has inside diameter 12 ft and height 80 ft without the lid. If the silo’s lid and curved walls
Use the formula obtained in Exercise 84 to find the slope of the level curve x2y + 2y3 − 2e−x = 14 at the point (0, 2). What is the equation of the tangent line to the level curve at this point?Data from Exercises 84Suppose y = h(x) is a differentiable function of x and that F(x, y) = C for
The likelihood that a person with a contagious disease will infect others in a social situation may be assumed to be a function f(s) of the distance s between individuals. Suppose contagious individuals are uniformly distributed throughout a rectangular region R in the xy plane. Then the likelihood
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.A rectangular garden that is 30 yards long and 40 yards wide is bordered by a concrete path that is 0.8 yard wide. Use calculus to estimate
A community is laid out as a rectangular grid in relation to two main streets that intersect at the city center. Each point in the community has coordinates (x, y) in this grid, for −10 ≤ x ≤ 10, −8 ≤ y ≤ 8 with x and y measured in miles. Suppose the value of the land located at the
Repeat Exercise 80 for V(x, y) = (300 + x + y)e−0.01x and the region −1 ≤ x ≤ 1, −1 ≤ y ≤ 1.Data from Exercises 80A community is laid out as a rectangular grid in relation to two main streets that intersect at the city center. Each point in the community has coordinates (x, y) in this
Repeat Exercise 80 for V(x, y) = 400xe−y and the region R: 0 ≤ y ≤ x, 0 ≤ x ≤ 1.Data from Exercises 80A community is laid out as a rectangular grid in relation to two main streets that intersect at the city center. Each point in the community has coordinates (x, y) in this grid, for −10
A health official wants to estimate the number of people susceptible to a new strain of influenza. Examinations conducted in the test region R shown in the accompanying figure suggest that the density of susceptible people in the region is p(x, y) = xy thousand people per square mile. How many
A box has the shape of the solid bounded above by the plane 3x + 4y + 2z = 12 below by the xy plane, and on the sides by the planes x = 0 and y = 0, where x, y, and z are in inches. Find the volume of the box.
A map of a small regional park is a rectangular grid, bounded by the lines x = 0, x = 4, y = 0, and y = 3, where units are in miles. It is found that the elevation above sea level at each point (x, y) in the park is given by E(x, y) = 90(2x + y2) feet Find the average elevation in the park.
A building is to have a curved roof above a rectangular base. In relation to a rectangular grid, the base is the rectangular region −30 ≤ x ≤ 30, −20 ≤ y ≤ 20, where x and y are measured in meters. The height of the roof above each point (x, y) in the base is given by h(x, y) = 12 −
In a psychological experiment, x units of stimulus A and y units of stimulus B are applied to a subject, whose performance on a certain task is then measured by the function P(x, y) = 10 + xye1−x2−y2 Suppose x varies between 0 and 1 while y varies between 0 and 3. What is the subject’s
Recall from Exercise 47, Section 7.1, that the surface area of a person’s body may be estimated by the empirical formula S(W, H) = 0.0072W0.425H0.725 where W is the person’s weight in kilograms, H is the person’s height in centimeters, and the surface area S is measured in square meters.a.
A storage bin is to be constructed in the shape of the solid bounded above by the surface z = 20 − x2 − y2 below by the xy plane, and on the sides by the plane y = 0 and the parabolic cylinder y = 4 − x2, where x, y, and z are in meters. Find the volume of the bin.
The population density is f(x, y) = 2,500e−0.01x−0.02y people per square mile at each point (x, y) within the triangular region R with vertices (−5, −2), (0, 3), and (5, −2). Find the total population in the region R.
A state legislature decides to create a new district from the region shown in the accompanying figure. It is found that the population density function for the region is p(x, y) = ye−0.2x thousand people per square mile.a. If c = 9, what is the total population in the region?b. The committee
The population density is f(x, y) = 1,000y2e−0.01x people per square mile at each point (x, y) within the region R bounded by the parabola x = y2 and the vertical line x = 4. Find the total population in the region R.
In Exercises 96 through 98, use double integration to find the required quantity. In some cases, you may need to use the numeric integration feature of your calculator.Find the area of the region bounded above by the curve (ellipse) 4x2 + 3y2 = 7 and below by the parabola y = x2.
In Exercises 29 through 58, factor the given polynomial using integer coefficients.2x2 − x − 15
In Exercises 29 through 58, factor the given polynomial using integer coefficients.12x2 − x − 20
In Exercises 96 through 98, use double integration to find the required quantity. In some cases, you may need to use the numeric integration feature of your calculator.Find the average value ofthe rectangular region bounded by the lines x = 1, x = 2, y = 1, and y = 3. f(x, y) = xy In (-) over
In Exercises 1 through 16, use L’Hôpital’s rule to evaluate the given limit if the limit is an indeterminate form. lim (1 + 2x)¹/x x→0
In Exercises 1 through 16, use L’Hôpital’s rule to evaluate the given limit if the limit is an indeterminate form. x³ - 3x² lim 4 x-0 3x² + 2x
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.) 1 (a")5 ,10 ||
In Exercises 96 through 98, use double integration to find the required quantity. In some cases, you may need to use the numeric integration feature of your calculator.Find the volume of the solid bounded above by the graph of f(x, y) = x2e−xy and below by the rectangular region R: 0 ≤ x ≤ 2,
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.) u_D/D 1 2 a D
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.)a2an = 1/a
In Exercises 37 through 42, factor the given polynomial using integral coefficients.4x2 + 12x + 9
In Exercises 29 through 58, factor the given polynomial using integer coefficients.x7 − x5
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.) n (a^)3 1 Va
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.)(a3)n = a12
In Exercises 37 through 42, factor the given polynomial using integral coefficients.12x2 + 5x − 3
In Exercises 29 through 58, factor the given polynomial using integer coefficients.x3 + 2x2 + x
In Exercises 43 through 76, simplify the given expression as much as possible. Assume a, b, and c are positive real numbers. 2.314 a²c² b
In Exercises 37 through 42, factor the given polynomial using integral coefficients.x3 + 3x2 − x − 3
In Exercises 29 through 58, factor the given polynomial using integer coefficients.2x3 − 8x2 − 10x
In Exercises 37 through 42, factor the given polynomial using integral coefficients.x4 − 5x2 + 4
In Exercises 29 through 58, factor the given polynomial using integer coefficients.x4 + 5x3 − 14x2
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