Questions and Answers of Business Economics And Finance

Convert the following exponential expressions to their equivalent logarithmic forms: (a) 64 = 82 (b) 81 = 34 (d) 7 = 491/2 (c) = 2-3 (e) 482/3 (f) 216 363/2 =
Change the following natural logarithms to their equivalent natural exponential forms: (a) In 132.56495 (d) In 624.12713 (b) In5.81.75786 (e) In y = 2.351 (f) Iny (c) In 0.4 -0.91629- 7-9t
Convert the following logarithms into their equivalent exponential forms: (a) log13 169 = 2 (d) logg1 9 = (b) logs 1253 (e) log4(4)=-3 (c) log6(6)=-2 (f) log 162 =
A village’s arable land L is eroding at a rate of 1.8 percent a year. If conditions continue, how much will be left in 15 years? L = Loe-0.018(15) Loe-0.27 L = 0.76338L0 76% of current arable land
A country’s timberland T is being cut back at a rate of 3.2 percent a year. How much will be left in 8 years?Using a calculator, T = Toe-0.032(8) Toe -0.256-
A country’s population P goes from 54 million in 1987 to 63.9 million in 1993. At what rate r is the population growing?Letting t = 0 for 1987 and t = 6 for 1993, and noting from the previous
A firm’s profits i’ have been growing consistently over time from $3.40 million in 1985 to $6.71 million in 1993. (a) Express profits as a natural exponential function of time π = π0ert and (b)
Given two sets of points from data growing consistently over time, such as subscriptions to a magazine numbering 6.25 million in 1988 and 11.1 million in 1993, (a) express subscriptions as a natural
Redo Problem 11.33 for a present value of $1500 to be paid in 8 years when the current interest rate is 5 percent. (a) (b) P=1500(1+.05)-8 P=1500(0.67684) = $1015.26 -4(8) .05 = P 1500 1+ =
Find the present value of $1000 to be paid 4 years from now when the current interest rate is 6 percent if interest is compounded (a) annually, (b) quarterly, and (c) continuously. (a) From (1116),
Discounting is the process of determining the present value P of a sum of money A to be received in the future. Find the formula for discounting under (a) annual compounding, (b) multiple
How long will it take money to treble at 12 percent interest compounded quarterly? From (11.8), .12 A-P(+)* 1+ 4 4(1) = P(1+.03)4
How many years t will it take a sum of money P to double at 6 percent interest compounded annually? A = P(1+.06)' For money to double, A = 2P. Substituting for A, Dividing by P, Taking the natural
Find the effective rate of interest under (a) semiannual, (b) quarterly, and (c) continuous compounding for Problem 11.26 where r was 9 percent. (a) From (11.14), (b) Substituting, (c) From (11.15),
Given a nominal rate of interest r = 8 percent, as in Problem 11.25, find the effective rate of interest under (a) semiannual, (b) quarterly, and (c) continuous compounding. Note that in (11.14)
Find the formula for finding the effective rate of interest re for multiple compoundings when t > 1. From the explanation of the effective rate of interest in Problem 11.24 (e), we can write mi "
Redo Problem 11.25, given P = $10, 000, r = 9 percent, and t = 3. (a) (b) (c) (d) A = 10, 000(1+.09) A = 10, 000(1.295029) = $12, 950.29 A = 10, 000 1+ .09 2 A = 10, 000(1+.045)6 2(3) A = 10,
Find the value A of a principal P = $3000 set out at an interest rate r = 8 percent for time t = 6 years when compounded (a) annually, (b) semiannually, (c) quarterly, and (d) continuously. (a) From
Find the value A of a principal P = $100 set out at an interest rate r = 12 percent for time t = 1 year when compounded (a) annually, (b) semiannually, (c) quarterly, and (d) continuously;
Combine rules to differentiate the following functions. (a) y = Inx = (lnx) By the generalized power function rule, (b) y = In 4x = (In 4x)2 y' = = 2(lnx) ( 2 ln x = x By the generalized power
Differentiate the following natural logarithmic functions, using the rule found in (11.6):Note how a multiplicative constant within the log expression drops out in differentiation as in
Combine rules to differentiate the following functions. (a) f(x)=7xex By the product rule, f'(x) = 7x (2ex) + ex (7) = 14xe2x+7e2x =7e2x (2x+1) (c) y = (e-4x)3 By the generalized power function rule,
Differentiate the following natural exponential functions, using the rule found in (11.5): d/dx[es(x)] = e8(x). g'(x). (a) f(x) = ex Let g(x)=6x, then g'(x) = 6, and from (11.5), f'(x) = ex. 6 = 6e6x
Using the techniques of Section 11.5, solve the following natural logarithmic functions for x: (a) 7 ln x 2.6 = 10 - Solving algebraically for Inx, 7 ln x = 12.6, In x = 1.8 Setting both sides of the
Use the techniques from Section 11.5 to solve the following natural exponential functions for x: (a) 7e3x = 630 Solving algebraically for e3x, 3x=90
Simplify each of the following exponential expressions: (a) e2in.x From (11.2), (b) e3nx+4lny e2ln.xeln.x2 = eln.x = x =X (c) e1/2 In 9x (d) e Inx-8 In y e3nx+4lnyelnxeln 4 = x3y4 = e1/2
Simplify the following natural logarithmic expressions: (a) In6+In.x In 6+Inx In 6.x (b) Inx-In x x7-In In x7 - In x = In x' = In.x5 = 5 ln x (c) In 12+ In 3 - In 4 12.3 In 12+In 3 In 4 In 1 = In 9 4
Simplify the following natural exponential expressions. (e4x)3 = 12x e e8y =3x+8y (b) (e4x)3 (a) ex (c) e3.x e7x e8 e4.x e2x (d) e7x e8x e4.x = = e7x-4x=e3x e2r e&x = e2r-8r = e-6x -13 =
Use the properties of exponents to simplify the following exponential expressions, assuminga, b > 0 and a ≠ b: ay =a 7x+9y (a) a*.a (b) a a = ax+y a7x.ay aSx a7x (c) ay ax (d) a5x = b.x =a5x-2y
Use the properties of logarithms to write the following natural logarithmic forms as sums, differences, or products: (a) In 49x6 In 49x6 In 49 + 6ln.x (c) In(x/y2) = (b) Inxy7 In x3y73 ln x+7 In y =
Use the properties of logarithms to write the following expressions as sums, differences, or products: (a) loga (23x) loga (23x)=loga 23 + loga x (c) logx4y5 log4y=4 loga +5 loga y (e) loga (3x/8y)
Solve the following for x, y, or a by finding the equivalent expression: (a) y=log20 400 400 = 20' y = 2 (c) log2 x = 4 x = 24 x = 16 (e) loga 49 = 2 49 = a a = 491/2 a = 7 (g) loga 9= 9 = a 2/3 a =
Change the following exponential forms to logarithmic forms.11.10 Convert the following natural exponential expressions into equivalent natural logarithmic forms: (a) 36=62 log6 36 = 2 log() = -2 (c)
Convert the following natural logarithms into natural exponential functions: (a) In 24=3.17805 24 = 3.17805 (c) In 443.78419 44=e 3.78419 (e) In y=-8x y = e8x (b) In 0.6 -0.51083 0.6 = e -0.51083 (d)
Change the following logarithms to their equivalent exponential forms: (a) log, 49=2 log749 (c) 49 = 72 log, () = -1 -1 (e) log648= = 9-1 12 8 = 641/2 (g) loga y=5x y=a5x (b) log4 64 = 3 64 = 43 (d)
Given (a) y = ex and (b) y = ln x, and using a calculator or tables, construct a schedule and draw a graph for each of the functions to show that one function is the mirror image or inverse of the
Construct a schedule and draw a graph for the following functions to show that one is the mirror image and hence the inverse of the other, noting that (1) the domain of (a) is the range of (b) and
Set up a schedule, rounding to two decimal places, for the following natural exponential functions y= ekx where k they are all negatively sloped and convex: y= (a) y = e-0.5.x (b) y = ex (c) y=e-2
Using a calculator or tables, set up a schedule for each of the following natural exponential functions-y = ekx where k > 0, noting (1) the functions never equal zero, (2) they all pass through
Make a schedule for each of the following exponential functions with 0 (a) (b) (c) x y x y x y -3 8 -3 -2 4 -2 29 27 -3 -2 465 64 16 -1 2 -1 3 -1 4 1 0 1 0 1 1 1 2 3 -2 -14 -100 See Fig. 11-4. 1 2
Make a schedule for each of the following exponential functions with base a > 1 and then graphthem on the same grid to convince yourself that (1) the functions never equal zero, (2) they all pass
Solve the following equations using the Gaussian elimination method: (a) 3x16x2 - 5x3 == (c) 4x17x2+2x3 = -6 -x18x29x3 = 93 12x29x384 -5x+6x22x3 = -122 (b) 5x18x2 + 2x3 = 26 -x1 7x2 - 4x3 = 6 +9x3 =
Use the Gaussian elimination method to solve each of the following systems of linear equations: (a) 4x + 7y = 131 8.x-3y=41 (c) 11x-4y = 256 3x+6y=6 (b) 6x+5y= 51 -x+8y = 124 (d) 13x+9y= 15 -
Give the dimensions of the following transpose matrices: (a) B′, (b) D′, (c) F′, and (d) H′.Refer to the following matrices in answering the questions below: 92 8 -5 A = B 3 4 7 6 -2 = [41] c
Find the following transpose matrices: (a) B′, (b) D′, (c) F′, and (d) H′.Refer to the following matrices in answering the questions below: 92 8 -5 A = B 3 4 7 6 -2 = [41] c = [ C 36 3 -8 13
In the matrices above, identify the following elements: (a) a21 (b) b12 (c) c23 (d) d12 (e) e31 (f) f12 (g)g13 (h) g32 (i) h23 (j) h31.Refer to the following matrices in answering the questions
Give the dimensions of (a) A, (b) C, (c) E, and (d) G.Refer to the following matrices in answering the questions below: 92 8 -5 A = B 3 4 7 6 -2 = [41] c = [ C 36 3 -8 13 D = 6 11 -5 15 -8 47
Redo Problem 5.34, given 2a. Multiply row 2 by : 1 0 I 27 man 267 0 1 45 0 -6 -1 -36 2b. Add 6 times row 2 to row 3 and ignore row 1 since a12 = 0. 3a. Multiply row 3 by 15. 0 8 0 1 45 43 267 45 18
Redo Problem 5.34, given 2x14x2+7x3 = 82 6x13x2 x3 = 11 x12x25x3 = -27 The augmented matrix is 2 4 7 82 [A|B] = 6 -3 1 11 1 2 -5-27 1a. Multiply row 1 by : 1 2 41 6 -3 1 11 1 2 -5 -27 1b. Subtract 6
Redo Problem 5.34, given 4x12x2+5x3 = 21 3x16x2 x3 = 31 x18x2+3x3 = 37 The augmented matrix is 42521 [A B] 3 6 1 31 1a. Multiply row 1 by: 1 8 3 37 3 6 1 31 1 8 3 37 1b. Subtract 3 times row 1 from
Redo Problem 5.34, given 6x1 + 4x2 = 47 2x1 +9x277
Redo Problem 5.34, given The augmented matrix is 4x1 +9x262 5x1+8x2 = 58 la. Multiply row 1 by: 1b. Subtract 5 times row 1 from row 2: 2a. Multiply row 2 by - 13: 4 9 62 [A|B]= 5 8 58 1 5 [ ] 8 58 [
Use the Gaussian elimination method to solve the following system of linear equations:First express the equations in an augmented matrix:Then apply row operations to convert the coefficient matrix on
Redo Problem 5.32, givenNote from Problems 5.32 and 5.33 that if the equations are arranged so that in each successive equation the same variables are always placed directly under each other, as
Express the following system of linear equations (a) in matrix form and (b) as an augmented matrix, letting A = the coefficient matrix, X = the column vector of variables, and B = the column vector
Find the product of the following matrices and their corresponding identity matrices, givenMultiplication of a matrix by a conformable identity matrix, regardless of the order of multiplication,
Use the inventory matrix for the company in Problem 5.3 and the price vector from Problem 5.15 to determine the value of the inventory in all four of the company’s outlets. V=QP. QP is defined: 4 x
Find BΑ from Problem 5.28. BA is defined: 2 1 = 1) x 3. BA will be 2 3. BA= 5 [619] -[3] 16 = RIC RC2 RC3 R2C1 R2C2 R2C3 5(6) 5(1) 5(9) 30 5 45 = = 2(6) 2(1) 2(9) 12 2 18
Find AB, givenAB is not defined: A = [619] B= 52
Find EF, given E = 825 F= [364] RICI RIC RC3] EF= R2C1 R2C2 R2C3 R3C1 R3C2 R3C3 EF is defined: 3 x (11) x 3. EF will be 3 x 3.
Find DC from Problem 5.25. DC is defined: 3 x 3 = 3 x 1. DC will be 3 x 1. DC == 2 7 41 591 .3 6 2 [RC] 2(3)+7(7)+4(5) 75 R2C1 = 5(3)+9(7)+1(5) = 83 == R3C1 3(3)+6(7)+2(5). .61
Find CD, given 3 2 7 4 C = 7 D = 59 1 5 3 6 2 CD is not defined: 3 x 13 x 3. Multiplication is impossible in the given order.
Find AB, given 7 1 9 A = [265] B = 4 36 5 8 2 AB is defined: 1 x (33) x 3. AB will be 1 x 3. AB = [RIC RC2 RC3] [2(7)+6(4)+5(5) 2(1) +6(3)+5(8) 2(9)+6(6)+5(2)] = [63 60 64]
Find EF, given 8 3 E = 9 2 4 651 F = 2 7 59 EF is defined: 2 x (33) x 2. EF will be 2 x 2. RICI RIC2 EF = R2C1 R2C2 = 9(8)+2(2)+4(5) 9(3)+2(7)+4(9) = 6(8)+5(2)+1(5) 6(3)+5(7)+1(9) 78 96 77 63 62 38
Find the product CD, given 4 1 5 8 6 C = 69 D = 4 3 1 7 2 CD is defined: 3 x (22) x 3. CD will be 3 x 3. RICI RIC RC3] CD R2C1 R2C2 R2C3 R3C1 R3C2 R3C3.
Redo Problem 5.17 for AB′ in Problem 5.19, where B′ is the transpose of B:Note from Problems 5.19 to 5.21 that AB ≠ BA ≠ AB′. This further reflects the fact that matrix multiplication is
Redo Problem 5.17 for BA in Problem 5.19. The product BA is defined: 3 x 2 = 2 x 2. BA will be 3 x 2. 8 BA= 6 7 4 2 3 5 = ] [ - 92 8(3) +1(9) = 6(3)+7(9) L4(3) + 2(9) RICI RIC2" R2C1 R2C2 R3C1 R3C2
Redo Problem 5.17, givenThe product AB is not defined:The matrices are not conformable in the given order. The number of columns (2) in A does not equal the’number of rows (3) in B. Hence the
Redo Problem 5.17, given 4 7 6 B = 523-819 A = The product AB is defined: 2 x 2 = 2 x 3. The product AB will be 2 3. RIC RC2 RC3 AB = R2C R2C2 R2C3 AB = 4(3)+7(8) 4(1)+7(2) 4(6)+7(9) 2(3)+5(8)
Determine whether the product AB is defined, indicate what the dimensions of the product matrix will be, and find the product matrix, givenThe product AB is defined:the product matrix will be 2 × 2.
Redo Problem 5.15 for outlet 3 in Problem 5.3.Here Q = [29 36 24 32], P remains the same, and V=QP=29(400) +36 (300) + 24(250) + 32(500) = 44,400
If the price of a TV is $400, the price of a stereo is $300, the price of a VCR is $250, and the price of a camcorder is $500, use vector multiplication to find the value of stock for outlet 2 in
Find AB, given 15 A [6 358] B = = 18 9 The product is defined: 1 x 4 = 4 x 1. AB = [6(11)+3(15) +5(18) +8(9)] = [273]
Find AB, given 21 A = [5 12] B = 10 The product AB is defined: 1 x 2 = 2 x 1. AB [5(21) +12(10)] = [225] =
Find AB, givenThe product AB is defined:The product will be a 1 × 1 matrix, derived by multiplying each element of the row vector A by its cbrresponding element in the column vector B, and then
A ski shop discounts all its skis, poles, and bindings by 25 percent at the end of the season.Assuming that V1 is the value of stock in its three branches prior to the discount, find the value V2
Find kA, given 7 -4 2 k = -3 A = -9 5 -6 1 -8 1 -3(7) -3(-4) -3(2) -21 12 -67 kA = -3(-9) -3(5) -3(-6) = 27-15 18 -3(1) -3(-8) -3(1) -3 24 -3
Determine Ak, givenHere k is a scalar, and scalar multiplication is possible with a matrix of any dimension. 16 1 A = 9 2 k=5 7 4
GivenDetermine for each of the following whether the products are defined, that is, conformable for multiplication in the order given. If so, indicate the dimensions of the product matrix (a) AB, (b)
A monthly report R on sales for the company in Problem 5.5 indicates: [21 16 36 18 44 26 21 19 R = 11 17 13 20 33 28 34 12. What is the inventory level 13 at the end of the month? 43 66 64 501 21 16
Find the difference A — B for each of the following: 6 3 7 (a) A = B 29 1 5 8 9 = [4 1 2] 6-5 3-8 7-9 -5 -2 A B = = 2-4 9-1 1-2 -2 8 -1 T 13 8 9 3 (b) A = B = = 18 15 5 7
The parent company in Problem 5.3 sends out deliveries to its stores:What is the new level of inventory? 8 6 9 5 4 7 752 D = = 63 308 5 9 9 7 4.
Find the sum A + B for the following matrices: 6 5 (a) A = B = 8 2 3 7 94
A company with four retail stores has 35 TVs t, 60 stereos s, 55 VCRs (videocassette recorders) v, and 45 camcorders c in store 1; 80t, 65s, 50v, and 38c in store 2; 29t, 36s, 24v, and 32c in store
Use your knowledge of subscripts and addresses to complete the following matrix, given a12 = 9, a21= −4, a13 = −5, a31 = 2, a23 = 7, and a32 = 3.Since the subscripts are always given in
(a) Give the dimensions of each of the following matrices, (b) Find their transposes and indicate the new dimensions.(a) Recalling that dimensions are always listed row by column or rc, A = 3 × 2, B
Use the elimination method to solve each of the followina svstems of simultaneous eauations: (a) 4x-3y = 22 7x-6y=34 (b) - 5x+8y = 42 15x+9y=6 (c) 24x7y = 37 -6x+9y=27 (d) 11x+3y = 53 == 4x 18y172 -
Use the substitution method to solve each of the following systems of simultaneous equations: (a) 3x - y = 1 (b) 7x+2y= 62 4x+6y=38 (d) 18x-2y= 32 12x+5y=-23 x+6y =26 (c) 5x+y= 26 8x-3y = 60
Find the break-even point for each of the following monopolistic firms, given (a) R(x) = x + 22x C(x)=7x+36 (c) R(x) = -2x + 85x == C(x) 11x+420 = (b) R(x)=-5x + 163x C(x)=23x + 800 (d) R(x)=-3x+201x
Find the break-even point for each of the following firms operating in a purely competitive market, given total revenue R(x) and total cost C(x) functions for each: (a) R(x) = 125.x C(x)=85x+5200 (b)
Solve algebraically each of the following systems of linear supply and demand equations expressed in the mathematical format Q = f(P): (a) Supply: Q = 25P-185 Demand: Q-32P + 1240 (c) Supply: Q = 42P
Using graphs on your own, with P on the vertical axis as in economics, find the equilibrium quantity and price (Qe, Pe) in each of the following markets: (a) Supply: P = 2+2 Demand: P = -Q+22 (c)
Use graphs on your own to solve each of the following systems of linear equations: (a) 7x+2y= 33 (b) 6x8y = 10 (c) 3x+4y = 26- 5x + 3y = 47 -x+9y=74 4x-9y=-42 (d) 6x-2y= 26 15x-5y = 85
Give the dimensions of the following systems of equations: (a) z4w+3x+6y - z=7w 2x 8y - z9w 4x+5y (b) y=6x+5x2 y = 4x1 +9x2 y = 3x1 +8x2 (c) y=8x+5 y=2x+9 y=3x+7
Given the domain restriction x ≥ 0, show (a) algebraically and (b) graphically that the inverse function exists for y = x2.(a) Given y = x2 and the restricted domain x ≥ 0, the inverse function
Test (a) algebraically and (b) graphically to see if the inverse function exists for y = x (4.83)
Given y = f(x), the inverse function is written x = f−1(y), which reads “x is an inverse function of y.”Find the inverse functions for the functions in Problem 4.27:To find an inverse function,
Given the implicit functionfind the explicit functions (a) y = g(x) and (b) x = h(y), if they exist.(a) To find the explicit function y = g(x), if it exists, solve the implicit function f(x, y)
Given the following linear supply and demand equations in standard form,(a) convert them to the slope-intercept form in a manner conformable to (1) economic modeling and (2) mathematical modeling;
Given the following linear supply and demand equations in standard form(a) convert them to the slope-intercept form in a manner conformable to (1) economic modeling and (2) mathematical modeling; (b)

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