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All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences
In many applications, it is necessary to write expressions in the form cxn, where c is a constant and n is an integer. In problems 49-56, write the expressions in this form. (x3)257E58E59E60EPresent value If an investment has a goal (future value) of $S after n years. Invested at interest rate i (as decimal), compounded annually then the present value P that must be invested is given by P=S(1+i)n . In problems 61 and 62, find P for the given S,nandi. 15,000after6yearsat11.5Present value If an investment has a goal (future value) of $S after n years. Invested at interest rate i (as decimal), compounded annually then the present value P that must be invested is given by P=S(1+i)n. In problems 61 and 62, find P for the given S,nandi. 80,000after20yearsat10.5Personal income The total U.S. personal income I (in billions of dollars) from 1960 and projected to 2024 can be approximated by I=533.6(1.065)t where t is the number of years after 1960 (Source: U.S. Bureau of Labor Statistics). What t -values correspond to the years 2000,2014 and 2024? What does the formula estimate the total U.S. personal income to be in 2000,2014 and 2024? The U.S. total personal income given by the U.S. Bureau of Labor Statistics for 2018 is 19,129.6 billion. What does the formula estimate it to be in 2018?64EDiabetes Using data from 2010 and projected to 2050, the number of millions of U.S. adults with diabetes, y, can be approximated by the formula y=1201+5.25(1.066)t where t is the number of years after 2000 (Source: Centers for Disease Control and Prevention). The actual numbers of millions given by the CDC for selected years are as follows. 20102015202032.337.350.0 For each of these years, find he number of U.S. adults with diabetes predicted by the formula. Round your answer to one decimal place. Using the number of U.S. adults with diabetes given above for 2015, how many more U.S. adults with diabetes does the formula predict will be added by 2025? Why is it reasonable that the formula approximating the number of U.S. adults with diabetes has an upper limit that cannot be exceeded? Use large t -values to discover this formula's upper limit.66EHealth care expenditures The national health care expenditure H (in billions of dollars) can be modeled (that is, accurately approximated) by the formula H=738.1(1.065)t where t is the number of years past 1990 (Source: U.S. Department of Health and Human Services). What t-value corresponds to 2000? Approximate the national health care expenditure in 2000. Approximate the national health care expenditure in 2010. Estimate the national health care expenditure in 2018.CHECK POINT Which of the following are not real numbers? (a) 643 (b) 64 (c) 0 (d) 14 (e) 15 (f) 18CHECK POINT (a) W rite as radicals x13,x25,x32. (b) Write with fractional exponents: x34=x?,1x=1x?=x?CHECK POINT Evaluate the following. (a) 823 (b) (8)23 (c) 823 (d) 823 (e) 7115Complete the following. (a) xx13x3=x? (b) x2x12=x? (c) (x23)3=x? (d) x32x12=x? (e) x32x=x? (f) (x4y2)32=?True or false. (a) 8x23x13=4x (b) (16x8y)34=12x6y34 (c) (x2y3)13=(y3x2)13=yx23CHECK POINT Simplify. 24x33y 12xy23x2y 24x4y33y27CP1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35EIn problems 31-36, simplify each expression. 32x5y37E38E39EIn problems 37-44, perform the indicated operations and simplify. 10xz1030x17zIn problems 37-44, perform the indicated operations and simplify. 12x3y1227xy2In problems 37-44, perform the indicated operations and simplify. 250xy7z418x17y2In problems 37-44, perform the indicated operations and simplify. 32a9b54162a174In problems 37-44, perform the indicated operations and simplify. 16x3y43128y23In problems 45-48, determine a value for x that makes each statement true. (A9)x=AIn problems 45-48, determine a value for x that makes each statement true. (B20)x=BIn problems 45-48, determine a value for x that makes each statement true. (R7)x=RIn problems 45-48, determine a value for x that makes each statement true. (T3)x=TIn problems 49-54, rationalize the denominator and then simpify. (T3)x=TIn problems 49-54, rationalize the denominator and then simpify. 58In problems 49-54, rationalize the denominator and then simpify. 51 m2xmx2.In problems 49-54, rationalize the denominator and then simpify. 5x3w4xw2In problems 49-54, rationalize the denominator and then simpify. m2x3mx53In problems 49-54, rationalize the denominator and then simpify. mx34y2z54In calculus, it is frequently important to write an expression in the form cxn, where c is a constant and n is a rational number. In problems 55-58, write each expression in this form. 23x23In calculus, it is frequently important to write an expression in the form cxn, where c is a constant and n is a rational number. In problems 55-58, write each expression in this form. 23x3457EIn calculus, it is frequently important to write an expression in the form cxn, where c is a constant and n is a rational number. In problems 55-58, write each expression in this form. xx359E60EIn calculus problems, the answers are frequently expected to be in a form with a radical instead of a fractional exponent. In problems 59-62 write each expression with radicals. 61. 12x1262ERichter scale the richter scale reading for an earth quake measures its intensity (as a multiple of some minimum intensity used for comparison)The intensity I corresponding to a Richter scale reading R is given by I=10R A quake measuring 8.5 on the richter would be severe. Express the intensity of such a quake in exponential form and in radical form. Find the intensity of a quake measuring 9.0. The San Francisco quake that occurred during the 1989 world series measured 6.9, and the March 2011 quake that devastated sendai, Japan, measured 9.0.Calculate the ratio of these intensities(larger to smaller).Sound intensity The intensty of sound I (as a multiple of the average minimum threshold of hearing intensity) is related to the decible level D ( or loudness of sound) according to I=10D10 (a) Express 10D10 using radical ntation. (b) The back ground noise level of a relatively quite room has a decibel reading of 32.Find the intensity I1 of this noise level. (c) Adecibel reading of 140 is at the threshold of pain if I2 is the intensity of this threshold and I1 is the intensity found in part (b), express the ratio I2I1 as a power of 10.Then approximate this ratio.65E66E67E68E69E70E71EAdvertising and sales Suppose it has been determined that the sales at Ewing Gallery decline after the end of an advertising campaign, with daily sales given by, S=2000(20.1x) where S is in dollars and x is the number of days after the campaign ends. What are the daily sales 10 days after the end of the campaign?73ERemove parenthesis and combine like terms: 9x5x(x+2)+4x22CP3CP1E2E3E4E5EThe expressions in Problems 5 and 6 are polynomials with the form anxn+an1xn1++a1x+a0, where n is a positive integer. Complete the following. For 5x34x17, (a)a3=? (b)a1=? (c)a2=? (d)17=a?7E8E9EIn problems 7-12, evaluate each algebraic expression at the indicated values of the variables. 3x24y22xy at x=3 and y=411E12E13E14E15E16E17EIn problems 15-22, simplify by combining like terms. (4rs2r2s11rs2)(11rs22rs+4r2s)In problems 15-22, simplify by combining like terms. [84(q+5)+q]In problems 15-22, simplify by combining like terms. x3+[3x(x33x)]In problems 15-22, simplify by combining like terms. x2[x(x21)+1(1x2)]+xIn problems 15-22, simplify by combining like terms. y3[y2(y3+y2)][y3+(1y2)]In problems 23-58, perform the indicated operations and simplify. (5x3)(7x2)In problems 23-58, perform the indicated operations and simplify. (3x2y)(2xy3)(4x2y2)In problems 23-58, perform the indicated operations and simplify. (39r3s2)(13r2s)In problems 23-58, perform the indicated operations and simplify. (15m3n)(5mn4)In problems 23-58, perform the indicated operations and simplify. ax2(2x2+ax+ab)In problems 23-58, perform the indicated operations and simplify. 3(3x2)In problems 23-58, perform the indicated operations and simplify. (3y+4)(2y3)In problems 23-58, perform the indicated operations and simplify. (4x1)(x3)In problems 23-58, perform the indicated operations and simplify. 6(12x2)(2x2)In problems 23-58, perform the indicated operations and simplify. 2(x3+3)(2x35)In problems 23-58, perform the indicated operations and simplify. (4x+3)2In problems 23-58, perform the indicated operations and simplify. (2y+5)2In problems 23-58, perform the indicated operations and simplify. (0.14x)(0.1+4x)In problems 23-58, perform the indicated operations and simplify. (x3y30.3)2In problems 23-58, perform the indicated operations and simplify. 9(2x+1)(2x1)In problems 23-58, perform the indicated operations and simplify. 3(5y+2)(5y2)In problems 23-58, perform the indicated operations and simplify. (x212)2In problems 23-58, perform the indicated operations and simplify. (23+x)(23x)In problems 23-58, perform the indicated operations and simplify. (0.1x2)(x+0.05)In problems 23-58, perform the indicated operations and simplify. (6.2x+4.1)(6.2x4.1)In problems 23-58, perform the indicated operations and simplify. (x2)(x2+2x+4)In problems 23-58, perform the indicated operations and simplify. (a+b)(a2ab+b2)In problems 23-58, perform the indicated operations and simplify. (x3+5x)(x52x3+5)In problems 23-58, perform the indicated operations and simplify. (x31)(x72x45x2+5)In problems 23-58, perform the indicated operations and simplify. (18m2n+6m3n+12m4n2)(6m2n)48EIn problems 23-58, perform the indicated operations and simplify. (24x8y4+15x5y6x7y)(9x5y2)In problems 23-58, perform the indicated operations and simplify. (27x2y218xy+9xy2)(6xy)51E52E53E54E55E56E57E58E59EIn problems 59 and 60, simplify each expression. (a)(2x3)(3x+2)(5x2)(x3) (b)2x3(3x+2)5x2(x3)61E62EIn Problems 61-68, perform the indicated operations with expressions involving fractional exponents and radicals and then simplify. (x12+1)(x122)64E65E66E67EIn Problems 61-68, perform the indicated operations with expressions involving fractional exponents and radicals and then simplify. (4x3)53[(4x3)83+3(4x3)53]69EProfit Suppose a companys revenue R (in dollars) from the sale of x units of its product is given by R=215x Suppose further that the total cost C (in dollars) of producing those x units is given by C=65x+15,000 If profit is revenue minus cost, find an expression for the profit from the production and sale of x units. Find the profit received if 1000 units are sold.71ECell phones Cell Pro makes cell phones and has weekly costs of 1500 for rent, utilities, and equipment plus labor and material costs of 18.50 for each phone it makes. If x represents the number of phones produced and sold, write an expression for Cell Pros weekly total cost. If Cell Pro sells the phones to dealers for 45.50 each, write an expression for the weekly total revenue for the phones. Cell Pros weekly profit is the total revenue minus the total cost. Write an expression for Cell Pros weekly profit.73EMedications Suppose a nurse needs 10 cc (cubic centimeters) of a 15.5 solution (that is, a solution that is 15.5 ingredient) of a certain medication, which must be obtained by mixing x cc of a 20 solution and y cc of a 5 solution. Write expressions involving x for y, the amount of 5 solution, The amount of ingredient in the x cc of 20 solution, The amount of ingredient in the 5 solution, The total amount of ingredient in the mixture.75EFactor the following. 8x312x 3x(x2+5)5(x2+5) x210x24 x25x+6 4x220x+25 10049x2Consider 10x217x20 andobserve that (10x2)(20)=200x2. Find two expressions whose product is 200x2 andwhose sum is 17x. Replace 17x in 10x217x20 with the two expressions in (a). Factor (b) by grouping.3CPIn Problems 1 and 2, factor out the common monomial factor. (a)9ab12a2b+18b2 (b)4x2+8xy2+2xy32E3E4E5E6E7E8E9E10E11E12E13E14EFactor each expression in Problems 7-18 as a product of binomials. 49a2144b216E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45EIn Problems 45-48, determine the missing factor. 2x14+4x34=2x14(?)47E48E49E50E51E52E53E54E55E56E57E58E59EVolume Squares of side x are cut from four corners of an 8-by-8-inch piece of cardboard, and an open-top box is formed (see the figure). The volume of the box is given by 64x32x2+4x3. Factor this expression.61EPower in a circuit Factor the following expression for the maximum power in a certain electrical circuit. (R+r)22r(R+r)Revenue Revenue R from the sale of x units of a product is found by multiplying the price by the number of items sold. Factor the right side of R=300xx2 What is the expression for the price of the item?64ESimplify: 2x24x2x2CP3CP4CPSimplify yx1yxxy6CPIn Problems 1-6 simplify each fraction. 18x3y39x3z2E3EIn Problems 1-6 simplify each fraction. x26x+8x2165E6E7E8E9E10E11E12E13E14EIn Problems 7-34, perform the indicated operations and simplify. 15ac27bd4a14b2d16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50EIn Problems 51 and 52, rationalize the numerator of each fraction and simplify. x+hxh52E53EAPPLICATIONS Focal length Two thin lenses with focal lengths p and q and separated by a distance d have their combined focal length given by the reciprocal of 1p+1qdpq Combine these fractions. Use the reciprocal of your answer in (a) to find the combined focal length.APPLICATIONS Average cost A companys average cost per unit when x units are produced is defined to be Averagecost=Totalcostx Use this equation in Problems 55 and 56. Suppose a companys average costs are given by Averagecost=4000x+55+0.1x Express the average-cost formula as a single fraction. Write the expression that gives the companys total costs.56E57E