Questions and Answers #5 Decimals

Questions and Answers Sheet 5 Decimals Question #1 Find any holes, vertical asymptotes, and horizontal asymptotes for the following rational function. 2(𝑥 − 3)(3𝑥 + 5) 𝑓(𝑥) = 3(3𝑥 + 4)(𝑥 − 3) The hole(s) is _____ The Vertical Asymptote(s) is _______ The Horizontal Asymptote is _______ Answer: Step 1 Given 𝑓(𝑥) = 2(𝑥−3)(3𝑥+5) 3(3𝑥+4)(𝑥−3) Step 2 Hole's accur when factors from the numerator and the denominator cancel. Here (x-3) is a common factor there will be a hole at the point where x-3=0 x=-3 The Hole's is x=-3 The Vertical asymptotes: Since 𝑓(𝑥) = 2(𝑥−3)(3𝑥+5) 3(3𝑥+4)(𝑥−3) To find vertical asymptote denominator (x) to zero and solve for x. so 3(3𝑥 + 4) = 0 −4 𝑥= 3 So vertical asymptote is 𝑥 = −4 3 Step 3 Horizontal asymptote: Since 𝑓(𝑥) = = 2(𝑥−3)(3𝑥+5) 3(3𝑥+4)(𝑥−3) 2(3𝑥 2 +5𝑥−9𝑥−15) 3(3𝑥 2 −9𝑥+4𝑥−12) 6𝑥 2 −3𝑥−30 𝑓(𝑥) = 9𝑥 2 −15𝑥−36 𝑎𝑥 𝑛 +.. If 𝑓(𝑥) = 𝑏𝑥 𝑚 +⋯ if n=m then the horizontal asymptote 𝑦 = 𝑎 𝑏 so here top degree = bottom degree 6 Then horizontal asymptote: 𝑦 = ⇒𝑦= 9 2 3 So horizontal asymptote: 𝑦 = 2 3 Question #2 A person buys a phone for $81 and signs up for a single-line phone plan with 2000 monthly anytime minutes. The plan costs $119.96 per month. Write an equation that can be used to determine the total cost, C(t), of this phone plan for t months. Then, find the cost for 22 months, assuming that the number of minutes the person uses does not exceed 2000 per month. An equation that can be used to determine the total cost, C(t), of the phone plan fot t months is C(t)=? Answer: Step 1 Refer to the question, As person buys a phone at $81 and monthly plan costs $ 119.96 Let t be the number of months and C(t) be the total cost of t months. t= number of months C(t)= total cost at t months Step 2 As the phone cost $86 which is fixed cost and monthly plan costs $ 119.96 is a variable cost. Then total cost is the of fixed cost and variable cost. Fixed cost =$81 Variable cost =$119.96(t) C(t)= Fixed cost + Variable cost =$81+$119.96(t) Step 3 Then to find the cost fro 22 months just plug the vale of t=22 in the equation of total cost as, C(t)=$81+$119.96(t) =$81+$119.96(22) =81+2639.12 =$2720.12 Question #3 The cost in dollars to produce x t-shirts is C(x)=3.56x+80. The revenue in dollars from sales of x t-shirts is R(x)=9.34x. a) Write and simplify a function P that gives profit in terms of x. b) Find the profit if 100 t-shirts are produced and sold. a) P(x)=? b) The profit if 100 t-shirts are produced and sold is $? Answer: a) P(x)=R(x)-C(x) =9.34x-3.56x-80 =5.78x-80 b) P(100)=5.78(100)-80 =578-80 =498 Question #4 A manufacturer of tools, selling rechargeable drills to a chain of home improvement stores, charges $2 more per drill than its manufacturing cost, m. The chain then sells each drill for 170% of the price that it paid the manufacturer. Find a function P(m) for th price at the home improvement stores. The function for the price at the home improvement stores is given by P(m)=? Answer: Step 1 Manufacturing cost is =$m Manufacturer charges $2 more per drill. So manufacturer sells at $(m+2) Step 2 The chain sells each drill at 170% of (m+2). P(m)=170% of (m+2) P(m)=1.70(m+2) P(m)=1.70m+3.40 Answer: 1.70m+3.40 Question #5 Write an algebraic equation for the following problem and then solve it. When Angela and Walker first started working for the supermarket, their weekly salaries totaled $500. Now during the last 25 years Walker has seen his weekly salary triple. Angela has seen her weekly salary become four times larger. Together their weekly salaries now total $1740. How much did they each make 25 years ago? The algebraic equation is ___ =1740. Answer: Step 1 Let the Angela and Walker weekly salaries 25 years ago in dollars be x and y. Since, 25 years ago their total weekly salaries is $500. ∴ 𝑥 + 𝑦 = 500 Step 2 Now, Angela and walker salaries are four and three times than earlier respectively. Since, their total weekly salaries is $1740. ∴ 4𝑥 + 3𝑦 = 1740 Step 3 Now, solving the equation From (i), we have x+y=500 x=500-y (ii) Step 4 Putting eq.(iii) in eq.(ii), we get 4(500-y)+3y=1740 2000-4y+3y=1740 -y=1740-2000 -y=-260 y=260 ∴ 𝑥 = 500 − 260 = 240 Step 5 Answer Therefore, Angela and Walker weekly salaries 25 years ago in dollars is 240 and 260 respectively. Question #6 The general solution of the differential equation 𝑦 𝑛 − 9𝑦 = 0 can be written in the form 𝑦(𝑥) = 𝐴𝑒 𝑚1 𝑥 + 𝐵𝑒 𝑚2 𝑥 , where A, B, 𝑚1 , 𝑚2 ∈ 𝑅 and 𝑚1 > 𝑚2 . a) Solve the auxiliary equation, and enter the values of 𝑚1 and 𝑚2 in the boxes. Do not enter decimals in your answers. Enter whole numbers, fractions or square roots as appropriate. If you need to enter a square root, e.g. √𝑥 , enter this as √(𝑥). 𝑚1 = _______ 𝑚2 = _______ Answer: Step 1 To Find We have to find the general solution of the differential equation y''-9y=0 and write it in the form of 𝑦(𝑥) = 𝐴𝑒 𝑚1𝑥 + 𝐵𝑒 𝑚2𝑥 . Step 2 Calculation y''-9y=0 Auxilary Equation: 𝑚2 − 9 = 0 𝑚 2 − 32 = 0 (𝑚 + 3)(𝑚 − 3) = 0 𝑚 = −3,3 Since roots of the auxiliary equation are distinct and -3 < 3, then the solution is 𝑦(𝑥) = 𝐴𝑒 3𝑥 + 𝐵𝑒 −3𝑥 Hence 𝑚1 = 3 and 𝑚2 = −3. Question #7 An electric utility company determines the monthly bill by charging 85.4 cents per kilowatt-hour (kWh) used plus a base charge of $17.55 per month. Thus, the monthly charge is given by the function f(W)=0.854W+17.55 dollars, where W is the number of kilowatt-hours. a. Find f (6000) and explain what it means. b. What is the monthly charge if 9500 kWh are used? a. Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answers. Type integers or decimals.) A. f(6000)= nothing. The charge for 6000 kWh is $nothing. B. f(6000)= nothing. The charge for nothing kWh is $6000. b. The monthly charge is $nothing if 9500 kWh are used. (Simplify your answer. Type an integer or a decimal.) Answer: Step 1 Given that An electric utility company determines the monthly bill by charging 85.4 cents per kilowatt-hour (kWh) used plus a base charge of $17.55 per month. Thus, the monthly charge is given by the function f(W)=0.854W+17.55 dollars, where W is the number of kilowatt-hours. Step 2 To find a. f(6000). b. What is the monthly charge if 9500 kWh is used. a. To find: f(6000). The monthly charge is given by function, f(W)=0.854W+17.55 So to find f(6000) substitute W=6000 in f(W)=0.854W+17.55, ⇒ 𝑓(6000) = 0.854(6000) + 17.55 ⇒ 𝑓(6000) = 5141.55 Therefore there is monthly charge of $5141.55 if 6000 kWh is used. Step 3 b. To find What is the monthly charge if 9500 kWh is used. Substitute 𝑊 = 9500 ∈ 𝑓(𝑊) = 0.854𝑊 + 17.55, 𝑓(9500) = 0.854(9500) + 17.55 ⇒ 𝑓(9500) = 8130.55 Therefore the monthly charge if 9500 kWh is used is $8130.55. Question #8 A pendulum swings 80cm on its first swing, 76cm on its second swing, 72.2cm on its third swing, and 68.59cm on its fourth swing. Complete parts a and b below. a. If the pattern continues, what explicit formula can be used to find the distance of the n-th swing? 𝑎𝑛 = 80(0.95)𝑛−1 (Simplify your answer. Use integers or decimals for any numbers in the expression.) b. Use your formula to find the distance of the 10-th swing. 𝑎10 =? 𝑐𝑚 (Type an integer or decimal rounded to two decimal places as needed.) Answer: Step 1 Given, a pendulum swings 80 cm on its first swing, 76 on its second swing, and 72.2 on its third swing, and 68.69 cm on its fourth swing. Step 2 Part (a), A pendulum swings as follows: First swing, 𝑎1 = 80𝑐𝑚 Second swing, 𝑎2 = 76𝑐𝑚 Third swing, 𝑎3 = 72.2𝑐𝑚 Fourth swing, 𝑎4 = 68.59𝑐𝑚 Common ratio, 𝑟1 = 𝑎2 𝑎1 = 76 80 = 0.95 𝑎3 72.2 = = 0.95 𝑎2 76 Since the common ratio is same, so the pendulum swings are in geometric sequence. First term, a=80 Common ratio, r=0.95 The n-th term of the geometric sequence is given by ∴ 𝑎𝑛 = 𝑎(𝑟)𝑛−1 ; where n= number of terms ⇒ 𝑎𝑛 = 80(0.95)𝑛−1 The distance of n-th swings is 𝑎𝑛 = 80(0.95)𝑛−1 . 𝑟2 = Step 3 Part(b), ∴ 𝑎𝑛 = 80(0.95)𝑛−1 Out n=10 , then ⇒ 𝑎10 = 80(0.95)10−1 ⇒ 𝑎10 = 80(0.95)9 ⇒ 𝑎10 = 50.4199 ⇒ 𝑎10 = 50.42𝑐𝑚 The distance of the 10-th swing is 𝑎10 = 50.42𝑐𝑚 Question #9 A roof has a 0.2-inch layer of ice on it from a previous storm. Another ice storm begins to deposit ice at a rate of 0.23 inches per hour. a) Find a formula for a linear function f that models the thickness of the ice on the roof x hours after the second ice storm started. b) How thick is the ice after 1.5 hours? a) f(x)=___ b) The ice is ___ inches thick after 1.5 hours. Answer: Step 1 Given data: The thickness of ice is c=0.2 in. The rate of deposition is m=0.23 in/hr . Step 2 Part (a) The standard linear function is given by, f(x)=mx+c The function of thickness can be given by, f(x)=0.23x+0.2 Here, x is the time in hour. Thus, the linear function is 𝑓(𝑥) = 0.23𝑥 + 0.2 . Step 3 Part (b) Given that the time is x=1.5 hr. Substitute value of x in linear function as calculated in part (a), y=0.23x+0.2 y=0.23 (1.5)+0.2 y=0.545 in Thus, the thickness of ice layer is 0.545 in after 1.5 hr. Question #10 A roof has a 0.4-inch layer of ice on it from a previous storm. Another ice storm begins to deposit ice at a rate of 0.26 inches per hour. a) Find a formula for a linear function f that models the thickness of the ice on the roof x hours after the second ice storm started. b) How thick is the ice after 3.5 hours? a) f(x)=___ b) The ice is ___ inches thick after 3.5 hours. Answer: Step 1 Given data: A roof has a 4 inch layer of ice on it Another ice storm begins to deposite ice at a rate of 26 inches per hour. Step 2 Part (a) To find a formula for a linear function f that models the thickness of the ice on the roof x hours after the second ice storm started. Here F(x)=.4+.26x Part (b) How thick is the ice after 3.5 hours. F(x)=.4+.26x Here x=3.5 hours 𝐹(3.5) = .4 + (26 × 3.5) =.4+.91 =1.31 The ice is 1.31 inches thick after 3.5 hours. Question #11 The total percent of individuals aged 16 to 24 enrolled in college as of October of each year who completed high school during the preceding 12 months is 𝑓(𝑥) = 66.925𝑥 0.043 , where x is the number of years after 1999. a) Use this model to estimate the total percent in 2003 and 2007. The total percent in 2003 is __% The total percent in 2007 is __% Answer: Step 1 Basic information Given function is 𝐹(𝑥) = 66.925𝑥 0.043 Where x starts from 1999 onwards. Step 2 Solution For year 2003 x=2003-1999=4 x=4 𝑓(4) = 66.925(4)0.043 =71.0357 ≅ 71.036 Simplify f For year 2007 x=2007-1999=8 x=8 𝑓(8) = 66.925(8)0.043 =73.1849 ≅ 73.185 Question #12 A woman's shoe that is size x in country A is size s(x) in country B, where s(x)=x-12 . A woman's shoe that is size x in country p(x)=x-1.8 . Find a function that will convert country A shoe size to country C shoe size. The expression for the function is ___. (Use integers or decimals for any numbers in the expression.) Answer: Step 1 If the shoe size in country A=x , then The shoe size in country B=s(x)=x-12 If the shoe size in country B=x , then The shoe size in country C=p(x)=x-1.8 Step 2 The shoe size in country A=x, then The shoe size in country B=s(x)=x-12=t(say) Then shoe size in country C=p(t)=t-1.8 Putting back the value of t, p(s(x))=x-12-1.8=x-5.8 Hence, if shoe size in country A is x, then shoe size in country B=x-13.8 Question #13 In 2000. 54% of the residents in a large city regularly used newspapers for getting news and this has decreased at an average rate of approximately 1.9% per year since then. Find a linear function in slopeintercept form that models this description. The function should model the percentage of residents, P(x), who regularly used the news outlet x years after 2000. P(x)= ____________ (Use integers or decimals for any numbers in the expression.) Answer: Step 1 Given, The percentage of residents who get news by reading newspapers is 54%. Average rate =1.9% per year Step 2 Let year 2000 as starting year, so x=0 and percentage of residents =54. Therefore, the coordinate of years and percentage of residents as (0, 54). The rate is decreasing by 1.9% So, the slope will be -1.9 Step 3 Formula used: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) Step 4 On putting the all values in the above formula, it gives y-54=-1.9(x-0) y-54=-1.9x y=-1.9x+54 Step 5 Answer: The function model the percentage of residents P(x) is equal to P(x)=-1.9x+54 Question #14 Write and algebraic equation for the following problem and then solve it. Creekside dormitory just purchased a new sofa on sale $456. The sale price was 60% of the original price. What was the original price of the sofa? Let x= original price of the sofa, Write an equation for the statement ''the sale price was 60% of the original price''. ___=456 (Use integers or decimals for any numbers in the expression.) The original price was $___. (Simplify your answer.) Answer: Step 1 Given sofa an sale =$456 sale price =60% of the original price Step 2 Let x= original price of the sofa ⇒ 60% 𝑜𝑓 𝑥 = 456 60𝑥 ⇒ = 456 100 ⇒ 60𝑥 = 45600 𝑥= 45600 60 ⇒ 𝑥 = $760 Original price of the sofa =$760 Question #15 Write an equal decimal having the given number of decimal places. Briefly describe the Decimal Squares for both decimals in the equation. a. 0.7= ______(hundredths) The Decimal Square for this decimal can be described by A. 7 out of 10 parts shaded, or 70 out of 100 parts shaded. B. 10 out of 7 parts shaded, or 100 out of 70 parts shaded. C. 70 out of 10 parts shaded, or 7 out of 100 parts shaded. Answer: Step 1 0.7 can be written as 7 10 7 10 = 70 100 means 7 out of 10 parts shaded Step 2 70 100 means 70 out of 100 parts shaded Option (A) Question #16 Grayson deposited $4,500 into a new account. - The account earned annual simple interest. - No withdrawals or deposits were made. - At the end of 2 years, the balance of the account was $5,265. What is the annual interest rate on this account? I=Prt, plug in numbers and solve equation in your calculator. Convert ALL answer choices to decimals (percent divided by 100) A. 4.7% B. 5.2% C. 7.3% D. 8.5% Answer: Grayson deposited $4.500. so P=$4500 At the end of 2 years, the balance of the account was $5265. So, interest earn in t=2 years, I=$5265-$4500 =$765 So, I=Prt gives 𝑟= 𝐼 𝑃𝑡 765 = (4500)(2) = 765 9000 = 0.085 = 0.085 × 100% =8.5% Answer: D. 8.5% Question #17 The following table shows the tuition for a semester at CBU in dollars for years since 1990. Year CBU Tuition 3 4440 6 5350 9 6570 12 7895 15 9315 18 11040 21 12760 A scatter plot for the data and a graph of a linear model in black and a quadratic model in red follows. 3691215182140005000600070008000900010000110001200013000 Use the above scatter plot to decide which model better fits the data. linear quadratic none Let C(t)C(t) be the cost of tuition at CBU in dollars for t years since 1990. A quadratic model for the data is C(t)C(t)= .Use three decimals in your answer. Estimate the tuition 2015. $ Use the model to predict the year in which the tuition will be $15925. Answer: Step 1 Here, the 𝑅2 value for quadratic model is more compared with the 𝑅2 value for linear model. Hence, quadratic model is preferred. Correct option: Quadratic 𝐶(𝑡) = 8.978𝑡 2 + 250.536𝑡 + 3578.571 Step 2 From the above scatter plot, the quadratic equation is, 𝐶(𝑡) = 8.978𝑡 2 + 250.536𝑡 + 3578.571 The estimate of Tuition fee in 2015: The value of t for 2015 is t=28 Substitute t=28 in the quadratic equation: 𝐶(28) = 8.978(28)2 + 25.236(28) + 3578.571 ≈ $11323.931 Step 3 Given C(t)=15925 . Therefore, 15925 = 8.978𝑡 2 + 25.236𝑡 + 3578.571 ⇒ 8.978𝑡 2 + 25.236𝑡 − 12346.428 = 0 ⇒ 𝑡 ≈ 36 (ignore the negative one) Question #18 The function that converts the dollars of Country A to the dollars of Country B according to January 13, 2008, values is f(x)=0.99286x, where x is the number of dollars of Country A and f(x) is the number of dollars of Country B. Complete parts a and b below. a. Find the inverse function for f. 𝑓 −1 (𝑥) = (Simplify your answer. Use integers or decimals for any numbers in the expression. Round to four decimal places as needed.) Answer: Step 1 Given: 𝑓(𝑥) = 0.99286𝑥 ⇒ (1), where x is the number of dollars of country A let 𝑦 = 𝑓(𝑥) ⇒ (2)is the number of dollars of country B therefore, 𝑥 = 𝑓 −1 (𝑦) Step 2 substitute Eq'n(2) in Eq'n(1) y=0.99288x 𝑦 𝑥= 0.99288 𝑓 −1 (𝑦) = 1.00717𝑦 Substitute x insted of y therefore, 𝑓 −1 (𝑥) = 1.00717𝑥 Question #19 The future value $5000 invested for 3 years at a rate r compounded annually is given by 𝑆 = 5000(1 + 𝑟)3 . Complete parts (a) through (d). a) Complete the table below to determine the future value of $5000 at certain interest rates. Rate 4% 5% 7.25% 10.5% Future Value $? $? $? $? (Type integers or decimal rounded to the nearest hundredth.) Answer: Step 1 Future value is calculate by using this formula: 𝐹𝑉 = 𝑃𝑉(1 + 𝑟)𝑛 r is interest rate and n is time period. Step 2 a) Present value =$5000 Time period =3 year - Future value at 4% interest rate 𝐹𝑉 = 𝑃𝑉(1 + 𝑟)𝑡 𝐹𝑉 = 5000(1 + 0.04)3 =$5624.32 - Future value at 5% interest rate 𝐹𝑉 = 5000(1 + 0.05)3 =$5788.13 - Future value at 7.25% interest rate 𝐹𝑉 = 5000(1 + 0.0725)3 =$6168.25 - Future value at 10% interest rate 𝐹𝑉 = 5000(1 + 0.10)3 =$6655 Question #20 Subtract the polynomials. (8𝑥 2𝑎 − 2𝑥 𝑎 + 0.2) − (𝑥 2𝑎 − 9𝑥 𝑎 − 0.2) = (Simplify your answer. Do not factor. Use integers or decimals for any numbers in the expression.) Answer: Step 1 The addition or subtraction of two algebraic expressions is performed by combining like terms (like terms are terms that contain the same variables raised to the same power). Step 2 To perform the subtraction, combine like terms (terms with the same power) and perform the subtraction or addition of each term to obtain the answer. (8𝑥 2𝑎 − 2𝑥 𝑎 + 0.2) − (𝑥 2𝑎 − 9𝑥 𝑎 − 0.2) = 8𝑥 2𝑎 − 2𝑥 𝑎 + 0.2 − 𝑥 2𝑎 + 9𝑥 𝑎 + 0.2 = (8𝑥 2𝑎 − 𝑥 2𝑎 ) + (−2𝑥 𝑎 + 9𝑥 𝑎 ) + (0.2 + 0.2) = 7𝑥 2𝑎 + 7𝑥 𝑎 + 0.4