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All Textbook Solutions for Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 1-66, find the derivative of the function. [HINT: See Quick Example 5-14.] g(x)=ex+exexex 56E In Exercises 1-66, find the derivative of the function. [HINT: See Quick Example 5-14.] g(x)=e3x1ex2ex 58E 59E In Exercises 1-66, find the derivative of the function. [HINT: See Quick Example 5-14.] f(x)=exxex In Exercises 1-66, find the derivative of the function. [HINT: See Quick Example 5-14.] f(x)=[ln(ex)]2ln[(ex)2]62E 63E 64E 65E 66E In Exercises 67-72, find the equation of the straight line described. Use graphing technology to check your answers by plotting the given curve together with the tangent line. Tangent to y=exlog2x at the point (1,0)In Exercises 67-72, find the equation of the straight line described. Use graphing technology to check your answers by plotting the given curve together with the tangent line. Tangent to y=ex+ex at the point (0,2)69E 70E In Exercises 67-72, find the equation of the straight line described. Use graphing technology to check your answers by plotting the given curve together with the tangent line. At right angles to ex2 at the point where x=1 72E In Exercises 73-78, use lHospitals rule to find the limits. [HINT: See Example 4.] limx+x+2ex In Exercises 73-78, use lHospitals rule to find the limits. [HINT: See Example 4.] limx+x2+x+1ex In Exercises 73-78, use lHospitals rule to find the limits. [HINT: See Example 4.] limx2x+3e2x 76E In Exercises 73-78, use lHospitals rule to find the limits. [HINT: See Example 4.] limx0ex1x 78E Research and Development: Industry The total spent on research and development by industry in the United States during 20022012 can be approximated by S(t)=29lnt+164billiondollars(2t12), where t is the year since 2000. What was the total spent in 2010(t=10), and how fast was it increasing? [HINT: See Quick Examples 1 and 2.]Research and Development: Federal The total spent on research and development by the federal government in the United States during 20022012 can be approximated by S(t)=3.1lnt+22billiondollars(2t12), where t is the year since 2000. What was the total spent in 2005(t=5), and how fast was it increasing? [HINT: See Quick Examples 1 and 2.]Research and Development: Industry (Refer to Exercise 79.) The function S(t) inExercise 79 can also be written (approximately) as S(t)=29ln(286t+2,860)billiondollars(8t2), wherethis time t is the year since 2010. Use this alternative formula to estimate the amount spent in 2010 and its rate of change, and check your answers by comparing them with those in Exercise 79.82E Carbon Dating The age in years of a specimen that originally contained 10 grams of carbon 14 is given by y=log0.999879(0.1x), where x is the amount of carbon 14 it currently contains Compute. dydx|x=5, and interpret your answer. [HINT: For the calculation, see Quick Examples 3 and 4.]84E 85E New York City Housing Costs: Uptown The average price of a two-bedroom apartment in uptown New York City during the real estate boom from 1994 to 2004 can be approximated by p(t)=0.14e0.10tmilliondollars(0t10), ` where t is time in years. (t=0 represents1994.) What was the average price of a two-bedroom apartment in uptown New York City in 2002, and how fast was the price increasing? (Round your answers to two significant digits.) [HINT: See Quick Example 14.]87E Big Brother The following chart shows the total number of wiretaps authorized each year by U.S. state courts from 1990 to 2013. (t=0 represents1990.)45 N(t) Wiretap orders, state counts only These data can be approximated by the model N(t)=410e0.071t(0t23) a. Find N(20)andN(20). Be sure to state the units of measurement. To how many significant digits should we round the answers? Why? b. The number of people whose communications are intercepted averages around 100 per wiretap order. What does the answer to part (a) tell you about the number of people whose communications were intercepted? c. According to the model, the number of wiretap orders each year (choose one) (A) increased at a linear rate (B) decreased at a quadratic rate (C) increased at an exponential rate (D) increased at a logarithmic rate over the period shown.Investments If $10,000 is invested in a savings account offering 4% per year, compounded continuously, how fast is the balance growing after 3 years? [HINT: See Quick Example 15.]90E Investments If $10,000 is invested in a savings account offering 4% per year, compounded semiannually, how fast is the balance growing after 3 years?Investments If $20,000 is invested in a savings account offering 3.5% per year, compounded semiannually, how fast is the balance growing after 3 years?93E The 2003 SARS Outbreak A few weeks into the deadly SARS (severe acute respiratory syndrome) epidemic in 2003, the number of cases was increasing by about 4% each day. On April 1, 2003, there were 1,804 cases. Find an exponential model that predicts the number A(t) ofpeople infected t days after April 1, 2003, and use it to estimate how fast the epidemic was spreading on April 30, 2003. (Round your answer to the nearest whole number of new cases per day.) [HINT: See Example 2.]95E 96E SAT Scores by Income The following bar graph shows U.S. math SAT scores as a function of household income.51 Family income ($1,000) a. Which of the following best models the data (C is a constant)? (A) S(x)=C133e0.0131x (B) S(x)=C+133e0.0131x (C) S(x)=C+133e0.0131x (D) S(x)=C133e0.0131x (S(x) is the average math SAT score of students whose household income in x thousand dollars per year.) b. Use S(x) topredict how a students math SAT score is affected by a 1,000 increasein parents income for a student whose parents earn $45,000. c. Does S(x) increasesor decreases as x increases? Interpret your answer.98E 99E 100E 101E Epidemics Another epidemic follows the curve P=2001+20,000e0.549tmillionpeople, where P is the number of people infected and t is in years. How fast is the epidemic growing after 10 years? After 20 years? After 30 years? (Round your answers to two significant digits.) [HINT: See Example 3.]103E 104E 105E Subprime Mortgage Debt during the Housing Bubble (Compare Exercise 104.) During the real estate run-up in 20002008 the value of subprime (normally classified as risky) mortgage debt outstanding in the United States could be approximated by A(t)=1,350x1+4.2(1.7)tpercent(0t8) t years after the start of 2000.58 a. How fast, to the nearest 1%.was the percentage increasing at the start of 2005? b. Compute limt+A(t) and limt+A(t). What do the answers tell you about subprime mortgages?107E 108E Radioactive Decay Plutonium 239 has a half-life of 24,400 years. How fast is a lump of 10 grams decaying after 100 years?110E 111E 112E Complete the following: The derivative of e raised to a glob is .114E 115E 116E What is wrong with the following? ddxln|3x+1|=3|3x+1| X WRONG!118E What is wrong with the following? ddx32x=(2x)32x1 X WRONG!120E 121E 122E 123E 124E 125E 126E In Exercises 110, find dy/dx, using implicit differentiation. In each case, compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative. [HINT: See Example 1.] 2x+3y=7 2E In Exercises 110, find dy/dx, using implicit differentiation. In each case, compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative. [HINT: See Example 1.] x22y=6 4E 5E 6E In Exercises 110, find dy/dx, using implicit differentiation. In each case, compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative. [HINT: See Example 1.] exy=1 In Exercises 110, find dy/dx, using implicit differentiation. In each case, compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative. [HINT: See Example 1.] exyy=2 9E 10E In Exercises 11-30, find the indicated derivative using implicit differentiation. [HINT: See Example 1.] x2+y2=5;dydx 12E 13E 14E In Exercises 11-30, find the indicated derivative using implicit differentiation. [HINT: See Example 1.] 3xyy3=2x;dydx 16E 17E 18E In Exercises 11-30, find the indicated derivative using implicit differentiation. [HINT: See Example 1.] p2pq=5p2q2;dpdq 20E In Exercises 11-30, find the indicated derivative using implicit differentiation. [HINT: See Example 1.] xeyyex=1;dydx 22E 23E In Exercises 11-30, find the indicated derivative using implicit differentiation. [HINT: See Example 1.] es2tst=1;dsst 25E 26E 27E 28E In Exercises 11-30, find the indicated derivative using implicit differentiation. [HINT: See Example 1.] ln(xy+y2)=ey;dydx 30E 31E 32E 33E In Exercises 3142, use implicit differentiation to find (a) the slope of the tangent line and (b) the equation of the tangent line at the indicated point on the graph. (Round answers to four decimal places as needed.) If only the x-coordinate is given, you must also find the y-coordinate. [HINT: See Example 2 and 3.] 2x2+xy=3y2,(1,1)In Exercises 3142, use implicit differentiation to find (a) the slope of the tangent line and (b) the equation of the tangent line at the indicated point on the graph. (Round answers to four decimal places as needed.) If only the x-coordinate is given, you must also find the y-coordinate. [HINT: See Example 2 and 3.] x2yy2+x=1,(1,0)36E 37E 38E In Exercises 3142, use implicit differentiation to find (a) the slope of the tangent line and (b) the equation of the tangent line at the indicated point on the graph. (Round answers to four decimal places as needed.) If only the x-coordinate is given, you must also find the y-coordinate. [HINT: See Example 2 and 3.] ln(x+y)x=3x2,x=0 40E In Exercises 3142, use implicit differentiation to find (a) the slope of the tangent line and (b) the equation of the tangent line at the indicated point on the graph. (Round answers to four decimal places as needed.) If only the x-coordinate is given, you must also find the y-coordinate. [HINT: See Example 2 and 3.] exyx=4x,x=3 42E 43E 44E In Exercises 43-52, use logarithmic differentiation to find dy/dx. Do not simplify the result. [HINT: See Example 4.] y=(3x+1)24x(2x1)3 46E In Exercises 43-52, use logarithmic differentiation to find dy/dx. Do not simplify the result. [HINT: See Example 4.] y=(8x1)1/3(x1)48E In Exercises 43-52, use logarithmic differentiation to find dy/dx. Do not simplify the result. [HINT: See Example 4.] y=(x3+x)x3+2 50E In Exercises 43-52, use logarithmic differentiation to find dy/dx. Do not simplify the result. [HINT: See Example 4.] y=xx 52E Productivity The number of CDs per hour that Snappy Hardware can manufacture at its plant is given by P=x0.6y0.4, Where x is the number of workers at the plant and y is the monthly budget (in dollars). Assume that P is constant, and compute dydx when x=100andy=200,000, Interpret the result. [HINT: See Example 5.]54E Demand The demand equation for soccer tournament T-shirts is xy2,000=y, Where y is the number of T-shirts the Enormous State University soccer team can sell at a price of $x per shirt. Find dydx|x=5, and interpret the result.56E 57E Employment An employment research company estimates that the value of a recent MBA graduate to an accounting company is V=3e2+5g3, where V is the value of the graduate, e is the number of year of prior business experience, and g is the graduate school grade-point average. If V is fixed at 200, find dedg when g=3.0, and interpret the result.Grades62 A productivity formula for a students performance on difficult English examination is g=4tx0.2t210x2(t30), where g is the score the student can expect to obtain, t is the number of hours of study for the examination, and x is the students grade-point average. a. For how long should a student with a 3.0 grade-point average study to score 80 on the examination? b. Find dtdx fora student who earns a score of 80, evaluate it when x=3.0, andinterpret the result.60E 61E 62E Fill in the missing terms: The equation x=y3+y3 specifies_______as a function of _____and _______as an implicit function of_________.64E Use logarithmic differentiation to give another proof of the product rule.66E 67E 68E 69E 70E In Exercises 18, find all the relative and absolute extrema of the given function on the given domain (if supplied) or on the largest possible domain (if no domain is supplied). f(x)=2x36x+1on[2,+]2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE Revenue Demand for the latest best-seller at OHaganBooks.com, A River Burns through It, is given by q=p2+33p+9(18p28) copiessold per week when the price is p dollars. What price should the company charge to obtain the largest revenue?24RE 25RE 26RE Office Space Although still a sophomore at college, John OHagans son Billy-Sean has already created several commercial video games and is currently working on his most ambitious project to date: a game called K that purports to be a simulation of the world. John OHagan has decided to set aside some office space for Billy-Sean against the northern wall in the headquarters penthouse. The construction of the partition will cost $8 per foot for the south wall and $12 per foot for the east and west walls. What are the dimensions of the office space with the largest area that can be provided for Billy-Sean with a budget of $480, and what is its area?28RE Box Design The sales department at OHaganBooks.com, which has decided to send chocolate lobsters to each of its best customers, is trying to design a shipping box with a square base. It has a roll of cardboard 36 inches wide from which to make the boxes. Each box will be obtained by cutting out corners from a rectangle of cardboard as shown in the following diagram: (Notice that the top and bottom of each box will be square, but the sides will not necessarily be square.) What are the dimensions of the boxes with the largest volume that can be made in this way? What is the maximum volume?30RE 31RE 32RE 33RE 34RE Elasticity of Demand (Compare Exercise 23.) Demand for the latest best-seller at OHaganBooks.com, A River Burns through It, is given by q=p2+33p+0(18p28) copiessold per week when the price is p dollars. a. Find the price elasticity of demand as a function of p. b. Find the elasticity of demand for this book at a price of $20 and at a price of $25. (Round your answers to two decimal places.) Interpret the answers. c. What price should the company charge to obtain the largest revenue?36RE 37RE 38RE In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 112, locate and classify all extrema in each graph. (By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum.) Also, locate any stationary points or singular points that are not relative extrema. [HINT: See the box titled Locating Candidates for Extrema.]In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=x24x+1withdomain[0,3]In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=2x22x+3withdomain[0,3]15E 16E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=t3+twithdomain[2,2]In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=2t33twithdomain[1,1]19E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] h(t)=t33t2withdomain[1,+] [HINT: See Example 2.]In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=x44x3withdomain[1,+]In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=3x43x3withdomain[1,+]In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] g(t)=14x423t3+12t2withdomain[,+]24E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] h(x)=(x1)2/3withdomain[0,2] [HINT: See Example 3.]26E 27E 28E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(t)=t2+1t21;2t2,t1 In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(t)=t21t2+1withdomain[2,2]In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=x(x1);x0 32E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] g(x)=x24x 34E 35E 36E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=xlnxwithdomain[0,+]38E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] g(t)=et-twithdomain[1,1]40E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=2x224x+4 42E In Exercises 13-44, find the exact location of all the relative and absolute extrema of the given function. [HINT: See Example 1.] f(x)=xe1x2 44E In Exercises 4548, use graphing technology and the method in Example 5 to find the x-coordinates of the critical points, accurate to two decimal places. Find all relative and absolute maxima and minima. [HINT: See Example 5.] y=x2+1x2withdomain(3,2)(2,6)46E 47E 48E In Exercises 4956 the graph of the derivative of a function f is shown. Determine the x-coordinates of all stationary and singular points of f, and classify each as a relative maximum, a relative minimum, or neither. (Assume that f(x) is defined and continuous everywhere in [3,3]) [HINT: See Example 5.]50E In Exercises 4956 the graph of the derivative of a function f is shown. Determine the x-coordinates of all stationary and singular points of f, and classify each as a relative maximum, a relative minimum, or neither. (Assume that f(x) is defined and continuous everywhere in [3,3]) [HINT: See Example 5.]52E In Exercises 4956 the graph of the derivative of a function f is shown. Determine the x-coordinates of all stationary and singular points of f, and classify each as a relative maximum, a relative minimum, or neither. (Assume that f(x) is defined and continuous everywhere in [3,3]) [HINT: See Example 5.]54E 55E 56E 57E 58E Draw the graph of a function that has stationary and singular points but no relative extrema.60E 61E 62E We said that if f is continuous on a closed interval [a,b], then it will have an absolute maximum and an absolute minimum. Draw the graph of a function with domain [0,1] havingan absolute maximum but no absolute minimum.64E 65E 66E In Exercises 1-8, solve the given optimization problems. [HINT: See Example 2.] Maximize P=xysubjecttox+y=10.In Exercises 1-8, solve the given optimization problems. [HINT: See Example 2.] Maximize P=xysubjecttox+2y=40.In Exercises 1-8, solve the given optimization problems. [HINT: See Example 2.] Minimize S=x+ysubjecttoxy=9 andboth xandy0.In Exercises 1-8, solve the given optimization problems. [HINT: See Example 2.] Minimize S=x+2ysubjecttoxy=2 andboth xandy0.In Exercises 1-8, solve the given optimization problems. [HINT: See Example 2.] Minimize F=x2+y2 subjectto x+2y=10.6E 7E 8E For a rectangle with perimeter 20 to have the largest area, what dimensions should it have?10E Advertising Costs The cost, in thousands of dollars, of airing x 30-second television commercials during a Super Bowl game can be approximated by1 C(x)=20+4,000x+0.05x2. How many 30-second television commercials should your company air to minimize average costs? What is the resulting average cost of a 30-second ad? [HINT: See Example 1.]Advertising Costs The cost, in billions of dollars, of airing x five-second hologram commercials during a Galactic Chess game can be approximated by C(x)=490+320x+0.001x2. How many hologram commercials should your company air to minimize average costs? What is the resulting average cost of a five-second ad? (Round your answer to the nearest billion dollars.) [HINT: See Example 1.]Average Cost: iPhones Assume that it costs Apple approximately C(x)=400,000+160x+0.001x2 dollarsto manufacture x 32GB iPhone 6s in an hour at the Foxconn Technology Group. How many iPhone 6s should be manufactured each hour to minimize average cost? What is the resulting average cost of an iPhone? How does the average cost compare with the marginal cost at the optimal production level? (Give your answer to the nearest dollar.)14E 15E 16E 17E 18E Fences I would like to create a rectangular vegetable patch. The fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. I have a budget of $80 for the project. What are the dimensions of the vegetable patch with the largest area I can enclose? [HINT: See Example 2.]Fences I would like to create a rectangular orchid garden that abuts my house so that the house itself forms the northern boundary. The fencing for the southern boundary costs $4 per foot, and the fencing for the east and west sides costs $2 per foot. If I have a budget of $80 for the project, what are the dimensions of the garden with the largest area I can enclose? [HINT: See Example 2.]21E 22E Fences (Compare Exercise 19.) For tax reasons I need to create a rectangular vegetable patch with an area of exactly 242 square feet. The fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. What are the dimensions of the vegetable patch with the least expensive fence? [HINT: Compare Exercise 3.]Fences (Compare Exercise 20.) For reasons too complicated to explain, I need to create a rectangular orchid gar- den with an area of exactly 324 square feet abutting my house so that the house itself forms the northern boundary. The fencing for the southern boundary costs $4 per foot, and the fencing for the east and west sides costs $2 per foot. What are the dimensions of the orchid garden with the least expensive fence? [HINT: Compare Exercise 4.]Revenue Hercules Films is deciding on the price of the video release of its film Son of Frankenstein. Its marketing people estimate that at a price of p dollars, it can sell a total of q=200,00010,000p copies. What price will bring in the greatest revenue? [HINT: See Example 3.]26E 27E Projected Revenue: Smartphones Worldwide annual sales of smartphones in 20132017 were projected to be approximately q=10p+4,360 millionphones at a selling price of $p per phone. What selling price would have resulted in the largest projected annual revenue? What would have been the resulting projected annual revenue?Revenue: Monorail Service The demand for monorail service in Las Vegas in 2005 can be approximated by q=4,500p+41,500 ridesper day when the fare was $p. What price should have been charged to maximize total daily revenue?6 30E 31E 32E 33E 34E Profit: Smartphones (Compare Exercise 27.) Worldwide annual sales of smartphones in 20122013 were approximately q=6p+3,030 millionphones at a selling price of $p per phone. Assuming a manufacturing cost of $80 per phone, what selling price would have resulted in the largest annual profit? What would have been the resulting annual profit? (The actual selling price in 2013 was $335.) [HINT: See Example 3, and recall that Profit=Revenue=Cost.]Projected Profit: Smartphones (Compare Exercise 28.) Worldwide annual sales of smartphones in 20132017 were projected at approximately q=10p+4,360 millionphones at a selling price of $p per phone.11 Assuming a manufacturing cost of $100 per phone, what selling price would have resulted in the largest projected annual profit? What would have been the resulting annual profit? [HINT: See Example 3, and recall that Profit=Revenue=Cost.]37E Profit Because of sales by a competing company, your companys sales of virtual reality video headsets have dropped, and your financial consultant revises the demand equation to p=800q0.35, where q is the total number of headsets that your company can sell in a week at a price of p dollars. The total manufacturing and shipping cost still amounts to $100 per headset. a. What is the greatest profit your company can make in a week, and how many headsets will your company sell at this level of profit? (Give answers to the nearest whole number.) b. How much, to the nearest $1, should your company charge per headset for the maximum profit?39E Metal Drums A company manufactures cylindrical metal drums with open tops with a volume of 1 cubic meter. What should be the dimensions of the drums in order to use the least amount of metal in their production? [HINT: See Example 4.]Tin Cans A company manufactures cylindrical tin cans with closed tops with a volume of 250 cubic centimeters. The metal used to manufacture the cans costs $0.01 per square centimeter for the sides and $0.02 per square centimeter for the (thicker) top and bottom. What should be the dimensions of the cans to minimize the cost of metal in their production? What is the ratio height/radius? [HINT: See Example 4.]42E 43E 44E Box Design A packaging company is going to make closed boxes, with square bases, that hold 125 cubic centimeters. What are the dimensions of the box that can be built with the least material?46E 47E Carry-on Dimensions American Airlines requires that the total outside (length+with+height) ofa carry-on bag not exceed 45 inches.13 Suppose you want to carry on a bag whose length is twice its height. What is the largest volume bag of this shape that you can carry on an American flight?49E 50E Package Dimensions The U.S. Postal Service (USPS) will accept packages only if the length plus girth is no more than 108 inches. (See the figure.) Assuming that the front face of the package (as shown in the figure) is square, what is the largest volume package that the USPS will accept?52E 53E Cellphone Revenues (Refer to Exercise 53.) If we assume instead that the revenue per cellphone user decreases continuously at an annual rate of 20%, we obtain the revenue model R(t)=350(39t+68)e0.2tmilliondollars. Determine (a) when to the nearest 0.1 year the revenue was projected to peak and (b) the revenue, to the nearest $1 million, at that time.

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