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mathematics
calculus

Questions and Answers of Calculus

A dieting company offers three foods, A, B, and C, and groups its customers into two groups according to their nutritional needs. The following table gives the percent of the daily nutritional
A hospital wishes to provide at least 24 units of nutrient A and 16 units of nutrient B in a meal, while minimizing the cost of the meal. If three types of food are available, with the nutritional
Three factories each dump waste water containing three different types of pollutants into a river. State regulations require the factories to treat their waste in order to reduce pollution levels.
Package Express delivers packages and overnight mailers. At each of several decentralized delivery depots in one district, Package Express hires a number of full- and part-time unskilled employees
Each nurse works 8 consecutive hours at the Beaver Medical Center. The center has the following staffing requirements for each 4-hour work period. Work Period Nurses Needed 1
Minimize g = 9y1 + 10y2 subject to y1 + 2y2 ≥ 21 3y1 + 2y2 ≥27 Complete the following. (a) Form the matrix associated with each given minimization problem and find its transpose. (b) Write the
Suppose a primal minimization problem and its dual maximization problem were solved by using the simplex method on the dual problem, and the final simplex matrix is given.(a) Find the solution of the
Minimize g = 2y1 + 10y2 subject to 2y1 + y2 ≥ 11 y1 + 3y2 ≥ 11 y1 + 4y2 ≥ 16 Write the dual maximization problem, and then solve both the primal and dual problems with the simplex method.
Minimize g = 8y1 + 4y2 subject to 3y1 + 2y2 ≥ 6 2y1 + y2 ≥11 Write the dual maximization problem, and then solve both the primal and dual problems with the simplex method.
Minimize g = 3y1 + y2 subject to 4y1 + y2 ≥ 11 3y1 + 2y2 ≥12 3y1 + y2 ≥ 6 Write the dual maximization problem, and then solve both the primal and dual problems with the simplex method.
Express each inequality as a ≤ constraint. 1. 3x - y ≥ 5 2. 4x - 3y ≥ 6 3. y ≥ 40 - 6x 4. x ≥ 60 - 8y
Use the simplex method to find the optimal solution. Assume all variables are nonnegative.1. Maximize f = 4x + y subject to5x + 2y ≤ 84- 3x + 2y ≥ 42. Maximize f = x + 2y subject to- x + 2y ≤
Use the simplex method or Excel or some other technology. Assume all variables are nonnegative.1. Minimize f = x + 2y + 3z subject tox + z ≤ 20x + y ≥ 30y + z ≤ 202. Maximize f = - x + 2y + 4z
A sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs
A cereal manufacturer makes two different kinds of cereal, Senior Citizen's Feast and Kids Go. Each pound of Senior Citizen's Feast requires 0.6 lb of wheat and 0.2 lb of vitamin-enriched syrup, and
A company manufactures commercial heating system components and domestic furnaces at its factories in Monaca, Pennsylvania, and Hamburg, New York. At the Monaca plant, no more than 1000 units per day
A manufacturer makes Portable Satellite Radios and Auto Satellite Radios at plants in Lakeland and Rockledge. At the Lakeland plant, at most 1800 radios can be produced, and the production of the
Nolan Industries manufactures water filters/purifiers that attach to a kitchen faucet. Each purifier consists of a housing unit that attaches to the faucet and a 60-day filter (sold separately) that
Johnson City Cooperage manufactures 30-gallon and 55-gallon fiber drums. Each 30-gallon drum takes 30 minutes to make, each 55-gallon drum takes 40 minutes to make, and the company has at most 10,000
Refer to Problem 29. Suppose that the cost of each commercial component is $380 at the Monaca plant and $400 at the Hamburg plant. The cost of each domestic furnace is $200 at the Monaca plant and
Refer to Problem 30. Suppose the costs associated with Portable Satellite Radios are $50 at Lakeland and $60 at Rockledge, and the costs associated with Auto Satellite Radios are $40 at Lakeland and
Three water purification facilities can handle at most 10 million gallons in a certain time period. Plant I leaves 20% of certain impurities, and costs $20,000 per million gallons. Plant II leaves
A chemical storage tank has a capacity of 200 tons. Currently the tank contains 50 tons of a mixture that has 10% of a certain active chemical and 1.8% of other inert ingredients. The owners of the
Manufacturing A ball manufacturer produces soccer balls, footballs, and volleyballs. The manager feels that restricting the types of balls produced could increase revenue. The following table gives
A farm co-op has two farms, one at Spring Run and one at Willow Bend, where it grows corn and soybeans. Differences between the farms affect the associated costs and yield of each crop, as shown in
Maximize f = 2x + 3y subject to 7x + 4y ≤ 28 - 3x + y ≥ 2 x ≥ 0, y ≥ 0 Complete both of the following. (a) State the given problem in a form from which the simplex matrix can be formed (that
Maximize f = 5x + 11y subject tox - 3y ≥ 3- x + y ≤ 1x ≤ 10x ≥ 0, y ≥ 0Complete both of the following.(a) State the given problem in a form from which the simplex matrix can be formed (that
A final simplex matrix for a minimization problem is given. In each case, find the solution.1.2.
Graph the solution set of each inequality or system of inequalities.1. 2x + 3y ‰¤ 122. 4x + 5y > 1003.4.
Maximize f = x + 4y subject to 7x + 3y ≤ 105 2x + 5y ≤ 59 x + 7y ≤ 70 Solve the linear programming problems using graphical methods. Restrict x ≥ 0 and y ≥ 0.
Maximize f = 2x + 5y subject to 4x + 5y ≤ 400 x ≤ 70 6x + 15y ≤ 900 Solve the linear programming problems using graphical methods. Restrict x ≥ 0 and y ≥ 0.
Minimize g = 5x + 3y subject to 3x + y ≥ 12 x + y ≥ 6 x + 6y ≥ 11 Solve the linear programming problems using graphical methods. Restrict x ≥ 0 and y ≥ 0.
Minimize g = x + 5y subject to 8x + y ≥ 85 x + y ≥ 50 x + 4y ≥ 80 x + 10y ≥ 104 Solve the linear programming problems using graphical methods. Restrict x ≥ 0 and y ≥ 0.
Maximize f = 5x + 2y subject to y ≤ 20 2x + y ≤ 32 - x + 2y ≥ 4 Solve the linear programming problems using graphical methods. Restrict x ≥ 0 and y ≥ 0.
Minimize f = x + 4y subject to y ≤ 30 3x + 2y ≥ 75 - 3x + 5y ≥ 30 Solve the linear programming problems using graphical methods. Restrict x ≥ 0 and y ≥ 0.
Maximize f = 7x + 12y subject to the conditions. 7x + 3y ≤ 105 2x + 5y ≤ 59 x + 7y ≤ 70 Use the simplex method to solve the linear programming problems. Assume all variables are nonnegative.
Maximize f = 3x + 4y subject to x + 4y ≤ 160 x + 2y ≤ 100 4x + 3y ≤ 300 Use the simplex method to solve the linear programming problems. Assume all variables are nonnegative.
Maximize f = 3x + 8y subject to the conditions. x + 4y ≤ 160 x + 2y ≤ 100 4x + 3y ≤ 300 Use the simplex method to solve the linear programming problems. Assume all variables are nonnegative.
Maximize f = 3x + 2y subject to x + 2y ≤ 48 x + y ≤ 30 2x + y ≤ 50 x + 10y ≤ 200 Use the simplex method to solve the linear programming problems. Assume all variables are nonnegative.
Maximize f = 4x + 4y subject to x + 5y ≤ 500 x + 2y ≤ 230 x + y ≤ 160 If there is no solution, indicate this; if there are multiple solutions, find two different solutions. Use the simplex
Maximize f = 2x + 5y subject to - 4x + y ≤ 40 x - 7y ≤ 70 If there is no solution, indicate this; if there are multiple solutions, find two different solutions. Use the simplex method, with x ≥
Minimize g = 7y1 + 6y2 subject to 5y1 + 2y2 ≥ 16 3y1 + 7y2 ≥ 27 Form the dual and use the simplex method to solve the minimization problem. Assume all variables are nonnegative.
Minimize g = 3y1 ≥ 4y2 subject to 3y1 + 5y2 ≥ 8 y1 + 5y2 ≥ 6 2y1 + 5y2 ≥ 18 Form the dual and use the simplex method to solve the minimization problem. Assume all variables are nonnegative.
Minimize g = 2y1 + y2 subject to the conditions. 3y1 + 5y2 ≥ 8 y1 + 5y2 ≥ 6 2y1 + 5y2 ≥ 18 Form the dual and use the simplex method to solve the minimization problem. Assume all variables are
Minimize g = 12y1 + 11y2 subject to y1 + y2 ≥ 100 2y1 + y2 ≥ 140 6y1 + 5y2 ≥ 580 Form the dual and use the simplex method to solve the minimization problem. Assume all variables are nonnegative.
Maximize f = 3x + 5y subject to x + y ≥ 19 - x + y ≥ 1 - x + 10y ≤ 190 Involve mixed constraints. Solve each with the simplex method. Assume all variables are nonnegative.
Maximize f = 4x + 6y subject to 2x + 5y ≤ 37 5x - y ≤ 34 - x + 2y ≥ 4 Involve mixed constraints. Solve each with the simplex method. Assume all variables are nonnegative.
Maximize f = 39x + 5y + 30z subject to 5x + 5z ≤ 7 3x + 5y ≤ 30 3x + 5y ≤ 18 Use the simplex method. Assume all variables are nonnegative.
Minimize g = 12y1 + 5y2 + 2y3 subject to y1 + 2y2 + y3 ≥ 60 12y1 + 4y2 + 3y3 ≥ 120 2y1 + 3y2 + y3 ≥ 80 Use the simplex method. Assume all variables are nonnegative.
Minimize g = 25y1 + 10y2 + 4y3 subject to 7.5y1 + 4.5y2 + 2y3 ≥ 650 6.5y1 + 3y2 + 1.5y3 ≥ 400 y1 + 1.5y2 + 0.5y3 ≥ 200 Use the simplex method. Assume all variables are nonnegative.
Minimize f = 10x + 3y subject to - x + 10y ≥ 5 4x + y ≥ 62 4x + y ≤ 50 Use the simplex method. Assume all variables are nonnegative.
Minimize f = 4x + 3y subject to - x + y ≥ 1 x + y ≤ 45 10x + y ≥ 45 Use the simplex method. Assume all variables are nonnegative.
Maximize f = 88x1 + 86x2 + 100x3 + 100x4 subject to 3x1 + 2x2 + 2x3 + 5x4 ≤ 200 2x1 + 2x2 + 4x3 + 5x4 ≤ 100 x1 + x2 + x3 + x4 ≤ 200 x1 ≤ 40 Use the simplex method. Assume all variables are
Maximize f = 8x1 + 8x2 + 12x3 + 14x4 subject to 6x1 + 3x2 + 2x3 + x4 ≤ 350 3x1 + 2x2 + 5x3 + 6x4 ≤ 300 8x1 + 3x2 + 2x3 + x4 ≤ 400 x1 + x2 + x3 + x4 ≤ 100 Use the simplex method. Assume all
Minimize g = 10x1 + 9x2 + 12x3 + 8x4 subject to 45x1 + 58.5x3 ≥ 4680 36x2 + 31.5x4 ≥ 4230 x1 + x2 ≤ 100 x3 + x4 ≤ 100 Use the simplex method. Assume all variables are nonnegative.
A company manufactures backyard swing sets of two different sizes. The larger set requires 5 hours of labor to complete, the smaller set requires 2 hours, and there are 700 hours of labor available
A company produces two different grades of steel, A and B, at two different factories, 1 and 2. The following table summarizes the production capabilities of the factories, the cost per day, and the
Chairco manufactures two types of chairs, standard and plush. Standard chairs require 2 hours to construct and finish, and plush chairs require 3 hours to construct and finish. Upholstering takes 1
A small industry produces two items, I and II. It operates at capacity and makes a profit of $18 on each item I and $12 on each item II. The following table gives the hours required to produce each
Pinocchio Crafts makes two types of wooden crafts: Jacob's ladders and locomotive engines. The manufacture of these crafts requires both carpentry and finishing. Each Jacob's ladder requires 1 hour
At its Jacksonville factory, Nolmaur Electronics manufactures 4 models of TV sets: LCD models in 27-, 32-, and 42-in. sizes and a 42-in. plasma model. The manufacturing and testing hours required for
A nutritionist wants to find the least expensive combination of two foods that meet minimum daily vitamin requirements, which are 5 units of A and 30 units of B. Each ounce of food I provides 2 units
A laboratory wishes to purchase two different feeds, A and B, for its animals. The following table summarizes the nutritional contents of the feeds, the required amounts of each ingredient, and the
A company makes three products, I, II, and III, at three different factories. At factory A, it can make 10 units of each product per day. At factory B, it can make 20 units of II and 20 units of III
A company makes pancake mix and cake mix. Each pound of pancake mix uses 0.6 lb of flour and 0.1 lb of shortening. Each pound of cake mix uses 0.4 lb of flour, 0.1 lb of shortening, and 0.4 lb of
A company manufactures desks and computer tables at plants in Texas and Louisiana. At the Texas plant, production costs are $36 for each desk and $60 for each computer table, and the plant can
Armstrong Industries makes two different grades of steel at its plants in Midland and Donora. Weekly demand is at least 500 tons for grade 1 steel and at least 450 tons for grade 2. Due to
A function and the graph of a feasible region are given. In each case, find both the maximum and minimum values of the function, if they exist, and the point at which each occurs.1. f = - x + 3y2. f
Maximize f = 5x + 6y subject to x + 3y ≤ 24 4x + 3y ≤42 2x + y ≤ 20 Solve the linear programming problems using graphical methods. Restrict x ≥ 0 and y ≥ 0.
In Problems 1-4, use a calculator to evaluate each expression. 1. (a) 100.5 (b) 52.7 2. (a) 103.6 (b) 8-2.6 3. (a) 3 1/3 (b) e2 4. (a) 211 6 (b) e-3
(a) Find a > 1 to express y = 3 (2/5) x in the form y = 3(a-x). (b) Are these functions growth exponentials or decay exponentials? Explain. (c) Check your result by graphing both functions.
(a) Find a > 1 to express y = 8 (5 / 7) x in the form y = 8(a-x). (b) Are these functions growth exponentials or decay exponentials? Explain. (c) Check your result by graphing both functions.
(a) Graph y = 2(1.5)-x. (b) Graph 2 (2 / 3)x. (c) Algebraically show why these graphs are identical.
(a) Graph y = 2.5(3.25)-x. (b) Graph y = 2.5 (4 / 13)x. (c) Algebraically show why these graphs are identical.
Given that y = (4/5)x, write an equivalent equation in the form y = b-x, with b > 1.
Given that y = 2.5-x, write an equivalent equation in the form y = bx, with 0 < b < 1.
In Problems 1-2, use a graphing utility to graph the functions.1. Given f(x) = e-x. Graph y = f (x) and y = f (kx) = e-kx for each k, where k = 0.1, 0.5, 2, and 5. Explain the effect that different
For Problems 1 and 2, let f (x) = c(1 + e-ax) with a > 0. Use a graphing utility to graph the functions.1. (a) Fix a = 1 and graph y = f(x) = c(1 + e-x) for c = 10, 50, and 100.(b) What effect
We will show in the next chapter that if $P is invested for n years at 10% compounded continuously, the future value of the investment is given by S = Pe0.1n Use P = 1000 and graph this function for
The percent concentration y of a certain drug in the bloodstream at any time t in minutes is given by the equation y = 100(1 - e-0.462t) Graph this equation for 0 ≤ t ≤ 10. Write a sentence that
A single bacterium splits into two bacteria every half hour, so the number of bacteria in a culture quadruples every hour. Thus the equation by which a colony of 10 bacteria multiplies in t hours is
A statistical study shows that the fraction of television sets of a certain brand that are still in service after x years is given by f(x) = e-0.15x. Graph this equation for 0 ≤ x ≤ 10. Write a
With U.S. Department of Health and Human Services data since 2000 and projected to 2018, the total public expenditures for health care H can be modeled by H = 624e0.07t where t is the number of years
Sketch the graph for t = 0 to t = 10 when the growth rate is 2% and N0 is 4.1 billion. World population can be considered as growing according to the equation N = N0(1 r)t where N0 is the number of
Sketch the graph for t = 0 to t = 10 when the growth rate is 3% and N0 is 4.1 billion. World population can be considered as growing according to the equation N = N0(1 r)t where N0 is the number of
Repeat Problem 37 when the growth rate is 5%. World population can be considered as growing according to the equation N = N0(1 r)t where N0 is the number of individuals at time t = 0, r is the yearly
Repeat Problem 38 when the growth rate is 7%. World population can be considered as growing according to the equation N = N0(1 r)t where N0 is the number of individuals at time t = 0, r is the yearly
With data from the U.S. Bureau of Labor Statistics for selected years from 1988 and projected to 2018, the billions of dollars spent for personal consumption in the United States can be modeled by P
Suppose that sales are related to advertising expenditures according to one of the following two models, where S1 and Sn are sales and x is advertising, all in millions of dollars.S1 = 30 + 20x -
The following table gives the millions of metric tons of carbon dioxide (CO2) emissions from biomass energy combustion in the United Sates for selected years from 2010 and projected to 2032.(a)
The following table gives the U.S. national debt for selected years from 1900 to 2013.(a) Using a function of the form y = a*b^x, with x = 0 in 1900 and y equal to the national debt in billions,
Total personal income in the United States (in billions of dollars) for selected years from 1960 and projected to 2018 follows.(a) These data can be modeled by an exponential function. Write the
Energy use per dollar of GDP indexed to 1980 means that energy use for any year is viewed as a percent of the use per dollar of GDP in 1980. The following data show the energy use per dollar of GDP,
The table below gives the U.S. consumer price index (CPI) for selected years from 2012 and projected to 2050. With the reference year as 2012, a 2020 CPI = 120.56 means goods and services that cost
The following table gives the value of an investment, after intervals ranging from 0 to 7 years, of $20,000 invested at 10%, compounded annually. (a) Develop an exponential model for these data,
As the baby boom generation ages and the proportion of the U.S population over age 65 increases, the number of Americans with Alzheimer's disease and other dementia is projected to grow each year.
In Problems 1-4, graph each function. 1. y = 4x 2. y = 8x 3. y = 2(3x) 4. y = 3(2x)
In Problems 1-3, use the definition of a logarithmic function to rewrite each equation in exponential form.1. 4 = log2 162. 4 = log3 813. 1 / 2 = log4 2
In Problems 1-4, write the equation in logarithmic form.1. 25 = 322. 53 = 1253. 4-1 = 1 / 44. 91/2 = 3
In Problems 1 and 2, write the equation in logarithmic form and solve for x. 1. e3x+5 = 0.55 (to 3 decimal places) 2. 102x + 1 = 0.25 (to 3 decimal places)

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