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mathematics
calculus

Mathematical Applications for the Management Life and Social Sciences 11th edition Ronald J. Harshbarger, James J. Reynolds - Solutions

If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball is given by S = 100 + 96t - 16t2 feet where t is the number of seconds after the ball is thrown. How long after it is thrown is the height 100 feet?
Solve each equation by factoring.1. x2 - 4x = 122. x2 = 11x = 103. 9 - 4x2 = 04. 25x2 - 16 = 0
A tennis ball is thrown into the air from the top of a hotel that is 350 feet above the ground. The height of the ball from the ground is given by D(t) = - 16t2 + 10t + 350 feet where t is the time, in seconds, after the ball is thrown. How long after the ball is thrown does it hit the ground?
The amount of airborne particulate pollution p from a power plant depends on the wind speed s, among other things, with the relationship between p and s approximated by p = 25 - 0.01s2 (a) Find the value(s) of s that will make p = 0. (b) What value of s from part (a) makes sense in the context of
The sensitivity S to a drug is related to the dosage size by S = 100x - x2 where x is the dosage size in milliliters. (a) What dosage(s) will yield 0 sensitivity? (b) Explain what your answer in part (a) might mean?
The time t, in seconds, that it takes a Corvette to accelerate to x mph can be described by t = 0.001(0.732x2 + 15.417x + 607.738) How fast is the Corvette going after 8.99 seconds? Give your answer to the nearest tenth.
Social Security benefits are paid from a trust fund. As mentioned in the Application Preview, the trust fund balance, B, in trillions of 2012 dollars, t years past the year 2000 is described by B = - 0.0046t2 - 0.033t + 6.05 Find in what year the trust fund balance is projected to be $5 trillion in
For the years from 2001 to 2012, the percent p of high school seniors who have tried marijuana can be considered as a function of time t according to p = f(t) = 0.17t2 - 2.61t + 52.64 where t is the number of years past 2000. In what year after 2000 is the percent predicted to reach 55%, if this
Two projectiles are shot into the air over a lake. The paths of the projectiles are given by(a) y = -0.0013x2 + x + 10 and(b)where y is the height and x is the horizontal distance traveled (both in feet). Determine which projectile travels farther by substituting y = 0 in each equation and finding
The Ace Jewelry Store sold a necklace for $144. If the percent profit (based on cost) equals the cost of the necklace to the store, how much did the store pay for it? Usewhere P is profit and C is cost?
The global spending on travel and tourism (in billions of dollars) can be described by the equation t = 0.787x2 - 11.0x + 290 where x equals the number of years past 1990. Find the year after 1990 in which spending is projected to reach $1000 billion?
Using data from 2006 and with projections to 2021, total annual expenditures for national health care (in billions of dollars) can be described by E = 7.94x2 + 33.2x + 2190 where x is the number of years past 2005. If the pattern indicated by this equation remains valid, when does the model predict
Because of friction from the walls of an artery, the velocity of a blood corpuscle in an artery is greatest at the center of the artery and decreases as the distance r from the center increases. The velocity of the blood in the artery can be modeled by the function v = k(R2 - r2) where R is the
The model for body-heat loss depends on the coefficient of convection K, which depends on wind speed v according to the equationK2 = 16v + 4where v is in miles per hour. Find the positive coefficient of convection when the wind speed is(a) 20 mph(b) 60 mph(c) What is the change in K for a change in
A fissure in the earth appeared after an earthquake. To measure its vertical depth, a stone was dropped into it, and the sound of the stone's impact was heard 3.9 seconds later. The distance (in feet) the stone fell is given by s = 16t12, and the distance (in feet) the sound traveled is given by s
(a) Find the vertex of the graph of the equation,(b) Determine if the vertex is a maximum or minimum point,(c) Determine what value of x gives the optimal value of the function, and(d) Determine the optimal (maximum or minimum) value of the function.1. y = ½ x2 + x2. y = x2 - 2x3. y = 8 + 2x -
For each function in Problems,(a) Tell how the graph of y = x2 is shifted, and(b) Graph the function.1. y = (x - 3)2 + 12. y = (x - 10)2 + 13. y = (x + 2)2 - 24. y = (x + 12)2 - 8
Graph each function with a graphing utility. Use the graph to find the vertex and zeros. Check your results algebraically?1.2. y = 0.1(x2 + 4x - 32)3. y = Â¼ x2 + 3x + 124. y = x2 - 2x + 5
Find the average rate of change of the function between the given values of x?1. y = - 5x - x2 between x = - 1 and x = 1.2. y = 8 + 3x + 0.5x2 between x = 2 and x = 4.
Find the vertex and zeros and use them to determine a range for a graphing utility that includes these values; graph the function with that range.1. y = 63 + 0.2x - 0.01x22. y = 0.2x2 + 16x + 1403. y = 0.0001x2 - 0.014. y = 0.01x - 0.001x2
(a) Find the vertex of each function f(x). Use the vertex to set the window in which to graph y = f(x) and then(b) Graphically approximate the solutions to f(x) = 0.1. f(x) = 8x2 - 16x - 162. f(x) = 3x2 - 18x + 16
Complete the following.(a) Use the graph of y = f(x) to find an integer solution to f(x) = 0.(b) Use the solution from (a) to find a factor of f (x).(c) Factor f (x).(d) Solve f (x) = 0.1. f(x) = 3x2 - 8x + 42. f(x) = 5x2 - 2x - 7
The daily profit from the sale of a product is given by P = 16x - 0.1x2 - 100 dollars.(a) What level of production maximizes profit?(b) What is the maximum possible profit?
The daily profit from the sale of x units of a product is P = 80x - 0.4x2 - 200 dollars. (a) What level of production maximizes profit? (b) What is the maximum possible profit?
1. The yield in bushels from a grove of orange trees is given by Y = x(800 - x), where x is the number of orange trees per acre. How many trees will maximize the yield?2. One of the early results in psychology relating the magnitude of a stimulus x to the magnitude of a response y is expressed by
The sensitivity S to a drug is related to the dosage x (in milligrams) by S = 1000x - x2 Sketch the graph of this function and determine what dosage gives maximum sensitivity. Use the graph to determine the maximum sensitivity?
If 100 feet of fence is used to enclose a rectangular yard, then the resulting area is given by A = x(50 - x) where x feet is the width of the rectangle and 50 - x feet is the length. Graph this equation and determine the length and width that give maximum area?
1. The rate of photosynthesis R for a certain plant depends on the intensity of light x, in lumens, according toR = 270x - 90x2Sketch the graph of this function, and determine the intensity that gives the maximum rate?2. A ball thrown vertically into the air has its height above ground given bys =
Two projectiles are shot into the air from the same location. The paths of the projectiles are parabolas and are given by(a) y = -0.0013x2 + x + 10 and(b)where x is the horizontal distance and y is the vertical distance, both in feet. Determine which projectile goes higher by locating the vertex of
The speed at which water travels in a pipe can be measured by directing the flow through an elbow and measuring the height to which it spurts out the top. If the elbow height is 10 cm, the equation relating the height h (in centimeters) of the water above the elbow and its velocity v (in
The following figure shows the graph of a total cost function, with x equal to the number of units produced.(a) Is the average rate of change of cost greater from x = a to x = b or from x = b to x = c? Explain.(b) Would the number of units d need to satisfy d
The following figure shows the graph of a total revenue function, with x equal to the number of units sold.(a) Is the average rate of change of revenue negative from x = a to x = b or from x = b to x = c? Explain.(b) Would the number of units d need to satisfy d
The owner of an apartment building can rent all 50 apartments if she charges $600 per month, but she rents one fewer apartment for each $20 increase in monthly rent.(a) Construct a table that gives the revenue generated if she charges $600, $620, and $640.(b) Does her revenue from the rental of the
The owner of a skating rink rents the rink for parties at $600 if 50 or fewer skaters attend, so that the cost per person is $12 if 50 attend. For each 5 skaters above 50, she reduces the price per skater by $0.50.(a) Construct a table that gives the revenue generated if 50, 60, and 70 skaters
The Pension Benefit Guaranty Corporation is the agency that insures pensions. The figure shows one study's projection for the agency's total resources, initially rising (from taking over the assets of failing plans) but then falling (as more workers retire and payouts increase).(a) What kind of
When a stone is thrown upward, it follows a parabolic path given by a form of the equation y = ax2 + bx + c If y = 0 represents ground level, find the equation of a stone that is thrown from ground level at x = 0 and lands on the ground 40 units away if the stone reaches a maximum height of 40
Rising health care costs are a growing concern for everyone in the United States. With U.S. Centers for Medicare and Medicaid Services data from 2006 and projected to 2021, U.S. per capita health care costs can be described by y = 20.61x2 - 116.4x + 7406 where x is the number of years past 2000.
Using U.S. Census Bureau data for selected years from 1970 and projected to 2040, the percent of the total workforce that is female is given by p(t) = -0.0036t2 + 0.38t + 38.62 where t is the number of years past 1970. (a) Graph the function y = p(t). (b) From the equation, identify the maximum
Online sales, which make up about 5% of total retail sales in the United States, are expected to top $400 billion by 2017. Using data from 2000 and projected to 2017, the revenue, in billions of dollars, can be described by y = 1.230x2 - 22.87x + 142.9 where x is the number of years past 1990. Use
Determine whether each function's vertex is a maximum point or a minimum point and find the coordinates of this point. Find the zeros, if any exist, and the y-intercept. Then sketch the graph of the function.1. y = x - ¼ x22. y = - 2x2 + 18x3. y = x2 + 4x + 44. y = x2 - 6x + 9
The total costs for a company are given by C(x) = 2000 + 40x + x2 and the total revenues are given by R(x) = 130x Find the break-even points?
If, in a monopoly market, the demand for a product is p = 1600 - x and the revenue is R = px, where x is the number of units sold, what price will maximize revenue?
1. The profit function for a certain commodity is P(x) = 110x - x2 - 1000. Find the level of production that yields maximum profit, and find the maximum profit?2. The profit function for a firm making widgets is P(x) = 88x - x2 - 1200. Find the number of units at which maximum profit is achieved,
(a) Graph the profit function P(x) = 80x - 0.1x2 - 7000. (b) Find the vertex of the graph. Is it a maximum point or a minimum point? (c) Is the average rate of change of this function from x = a < 400 to x = 400 positive or negative? (d) Is the average rate of change of this function from x = 400
(a) Graph the profit function P(x) = 50x - 0.2x2 - 2000. (b) Find the vertex of the graph. Is it a maximum point or a minimum point? (c) Is the average rate of change of this function from x = a < 125 to x = 125 positive or negative? (d) Is the average rate of change of this function from x = 125
(a) Form the profit function for the cost and revenue functions in Problem 3, and find the maximum profit.In problem 3,If a company has total costs C(x) = 15,000 + 35x + 0.1x2 and total revenues given by R(x) = 385x - 0.9x2, find the break-even points.(b) Compare the level of production to maximize
(a) Form the profit function for the cost and revenue functions in Problem 4, and find the maximum profit.In problem 4,If total costs are C(x) = 1600 + 1500x and total revenues are R(x) = 1600x - x2, find the break-even points.(b) Compare the level of production to maximize profit with the level to
Suppose a company has fixed costs of $28,000 and variable cost per unit of 2/5 x + 222 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is 1250 - 3/5 x dollars per unit.(a) Find the break-even points.(b) Find the maximum revenue.(c) Form
Suppose a company has fixed costs of $300 and variable cost per unit of 3/4x + 1460 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is 1500 - 1/4 x dollars per unit.(a) Find the break-even points.(b) Find the maximum revenue.(c) Form the
The following table gives the total revenues of Cable-net Communications for selected years.Year ______________ Total Revenues (millions)2008 ............................... $63.132009 ................................ 62.912010 ................................ 60.532011
If a firm has the following cost and revenue functions, find the break-even points?
Assume that sales revenues for Continental Divide Mining can be modeled byR(t) = 0.031t2 + 0.776t + 0.179where t is the number of years past 2003.(a) Use the function to determine the year in which maximum revenue occurs and the maximum revenue it predicts.(b) Check the result from (a) against the
Assume that costs and expenses for Continental Divide Mining can be modeled byC(t) = - 0.012t2 + 0.492t + 0.725where t is the number of years past 2003.(a) Use R(t) as given in Problem 20 and form the profit function (as a function of time).(b) Use the function from (a) to find the year in which
A supply function and a demand function are given.(a) Sketch the first-quadrant portions of those functions on the same set of axes.(b) Label the market equilibrium point.(c) Algebraically determine the market equilibrium point.1. Supply: p = q2 + 8q + 16Demand: p = 216 - 2q2. Supply: p = ¼ q2 +
If the supply function for a commodity is p = q2 + 8q + 16 and the demand function is p = - 3q2 + 6q + 436, find the equilibrium quantity and equilibrium price?
If the supply function for a commodity is p = q2 + 8q + 20 and the demand function is p = 100 - 4q - q2, find the equilibrium quantity and equilibrium price?
If the demand function for a commodity is given by the equation p2 + 4q = 1600 and the supply function is given by the equation 300 -p2 + 2q = 0, find the equilibrium quantity and equilibrium price?
If the supply and demand functions for a commodity are given by 4p - q = 42 and (p + 2)q = 2100, respectively, find the price that will result in market equilibrium?
If the supply and demand functions for a commodity are given by p - q = 10 and q(2p - 10) = 2100, what is the equilibrium price and what is the corresponding number of units supplied and demanded?
If a company has total costs C(x) = 15,000 + 35x + 0.1x2 and total revenues given by R(x) = 385x - 0.9x2, find the break-even points?
If the supply and demand functions for a certain product are given by the equations 2p - q + 6 = 0 and (p + q) (q + 10) = 3696, respectively, find the price and quantity that give market equilibrium?
The supply function for a product is 2p - q - 10 = 0, while the demand function for the same product is (p + 10)(q + 30) = 7200. Find the market equilibrium point?
The supply and demand for a product are given by 2p - q = 50 and pq = 100 + 20q, respectively. Find the market equilibrium point?
For the product in Problem 31, if a $22 tax is placed on production of the item, then the supplier passes this tax on by adding $22 to his selling price. Find the new equilibrium point for this product when the tax is passed on. (The new supply function is given by p = 1/2q + 27.) In problem
For the product in Problem 32, if a $12.50 tax is placed on production and passed through by the supplier, find the new equilibrium point. In problem 32, The supply and demand for a product are given by 2p - q = 50 and pq = 100 + 20q, respectively. Find the market equilibrium point?
If total costs are C(x) = 1600 + 1500x and total revenues are R(x) = 1600x - x2, find the break-even points?
Given that P(x) = 11.5x - 0.1x2 - 150 and that production is restricted to fewer than 75 units, find the break-even points?
If the profit function for a firm is given by P(x) = -1100 + 120x - x2 and limitations on space require that production is less than 100 units, find the breakeven points?
Find the maximum revenue for the revenue function R(x) = 385x - 0.9x2?
1. Find the maximum revenue for the revenue function R(x) = 1600x - x2?2. If, in a monopoly market, the demand for a product is p = 175 - 0.50x and the revenue function is R = px, where x is the number of units sold, what price will maximize revenue?
Decide whether each function whose graph is shown is the graph of a cubic (third degree) or quartic (fourth-degree) function.1.(a)(b)2.(a)(b)Match each equation with the correct graph among those labeled (a)-(h) by recognizing shapes and features of polynomial and rational functions. Use a graphing
Graph the functions given below 1. y = (x + 1) (x - 3) (x - 1) 2. y = x3 - 8x2 + 19x - 12 3. y = 1 - 2x / x 4. y = x / x - 1
(a) Graph each function with a graphing utility;(b) Classify each function as a polynomial function, a rational function, or a piecewise defined function;(c) Identify any asymptotes; and(d) Use the graphs to locate turning points.1. f(x) = 1.6x2 - 0.1x42. f(x) = x4 - 4x3 / 33. f(x) = 2x + 4 / x +
If a box with a square cross section is to be sent by the postal service, there are restrictions on its size such that its volume is limited by V = x2(108 - 4x), where x is the length of each side of the cross section (in inches).(a) If V = V(x), find V (10) and V (20).(b) What restrictions must be
Fixed costs FC are business costs that remain constant regardless of the number of units produced. Some items that might contribute to fixed costs are rent and utilities. FC = 2000 Is an equation indicating that a business has fixed costs of $2000. Graph FC = 2000 by putting x (the number of units
Global spending on travel and tourism (in billions of dollars) can be modeled by f(x) = 15.875x1.18 where x is the number of years after 1980. (a) Does the graph of this function bend upward or bend downward? (b) Graph this function for the years 1980-2020. (c) Use the model to find when global
For fiddler crabs, data gathered by Thompson show that the relationship between the weight C of the claw and the weight W of the body is given by C = 0.11W1.54 where W and C are in grams. (a) Does the graph of this function bend upward or bend downward? (b) Graph this function. (c) Use this
Suppose that the cost C (in dollars) of removing p percent of the particulate pollution from the smokestacks of an industrial plant is given by(a) Is C(p) undefined at any p-value? If so, what value? (b) What is the domain of C(p) as given by the equation? (c) What is the domain of C(p) in the
If the weekly total cost of producing 27" Toshiba television sets is given by C(x) = 50,000 + 105x, where x is the number of sets produced per week, then the average cost per unit is given by(a) What is the average cost per set if 30000 sets are sold? (b) Graph this function. (c) Does the average
If 100 feet of fence is to be used to enclose a rectangular yard, then the resulting area of the fenced yard is given by A = x(50 - x) Where x is the width of the rectangle. (a) If A = A(x), find A(2) and A(30). (b) What restrictions must be placed on x (the domain) so that the problem makes
The monthly charge for water in a small town is given byWhere x is water usage in hundreds of gallons and f(x) is in dollars. Find the monthly charge for each of the following usages: (a) 30 gallons, (b) 3000 gallons, and (c) 4000 gallons, (d) Graph the function for 0 ( x (100?
By using data for selected years from 1950 and projected to 2040, the gross domestic product (GDP), in billions of dollars, of the United States is modeled byWhere x is the number of years past 1950 (Sources: Bureau of Economic Analysis; U.S. Energy Information Administration). (a) Graph the
The monthly charge (in dollars) for x kilowatt hours (kWh) of electricity used by a commercial customer is given by the following function.Find the monthly charges for the following usages. (a) 5 kWh (b) 6 kWh (c) 3000 kWh
The postage charged for first-class mail is a function of its weight. The U.S. Postal Service uses the following table to describe the rates. Weight Increment ______________________ Rate ($) First ounce or fraction of an ounce .................. 0.49 Each additional ounce or fraction
In a given year the U.S. federal income tax owed by a married couple filing jointly can be found from the following table:(a) For incomes up to $137,300, write the piecewise defined function T with input x that models the federal tax dollars due as a function of x, the taxable income dollars
The demand function for a product is given by P = 200 / 2 + 0.1x Where x is the number of units and p is the price in dollars. (a) Graph this demand function for 0 ( x ( 250, with x on the horizontal axis. (b) Does the demand ever reach 0?
In studying lynchings between 1899 and 1946, psychologist Brian Mullin found that the size of a lynch mob relative to the number of victims predicted the level of brutality. He developed a formula for the other-total ratio (y) that predicts the level of self-attentiveness of people in a crowd of
A manufacturer estimates that the cost of a production run for a product isWhere x is the number of machines used. (a) Graph this total cost function for values of x ( 0. (b) Interpret any turning points. (c) Interpret the y-intercept?
Determine whether the scatter plot should be modeled by a linear, power, quadratic, cubic, or quartic function.1.2. 3. 4. 5. 6. 7. 8.
(a) Plot the given points below(b) Determine what type of function best models the data, and(c) Find the equation that is the best fit for the data.1.2. 3. 4.
The following table gives the projected population, in thousands, of U.S. females under the age of 18.(a) Create a linear function that models the projected population y, in thousands, as a function of the number of years past 2010. (b) What is the projected population of females under age 18 in
The following table gives the millions of metric tons of carbon dioxide emissions in the United States for selected years from 2010 and projected to 2032.(a) Create a linear function that models these data, with x as the number of years past 2010 and y as the millions of metric tons of carbon
Disposable income is the amount left after taxes have been paid and is one measure of the health of the economy. The following table gives the total U.S disposable income (in billions of 2005 dollars) for selected years from 2010 and projected to 2040.(a) Use x as the number of years past 2010 and
As the following table shows, projections indicate that the percent of U.S. adults with diabetes could dramatically increase.(a) Find a linear model that fits the data in the table, with x = 0 in 2000.(b) Use the model to predict the percent of U.S. adults with diabetes in 2018.(c) In what year
The table gives the wind chill temperature when the outside temperature is 20°F.(a) Use x as the wind speed and create a quadratic model for these data. (b) At what wind speed does the model predict that the wind chill temperature will be lowest? (c) Do you think the model found in part (a) is
The developing economies' share of the global gross domestic product (GDP) from 2003 and projected to 2017 is shown in the following table.(a) Find the quadratic function that best models the developing economies' share of the global GDP as a function of the number of years after 2000.(b) Use
The table gives the gross domestic product (GDP), in billions of dollars, of the United States for selected years from 2000 and projected to 2070.(a) Create a scatter plot of the data, with y representing GDP in billions of dollars and x representing the number of years past 2000.(b) Find the
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. The table below gives the millions of U.S. citizens age 65 and older with Alzheimer's from 2000 and
The following table gives the U.S. crude oil production, in billions of barrels, for selected years from 2010 and projected to 2030.(a) Find the linear function that models the data, with x equal to the number of years past 2010 and y equal to the number of billions of barrels of crude oil.(b) Find
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, as a percent, for selected years from 1985 and projected to 2035.(a) Find a linear function that

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