Questions and Answers #6 Decimals

Questions and Answers Sheet 6 Decimals Question #1 Given that a stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3.4 ft/sec. Complete parts a through c. a) Find a function for the radius in terms of t. r(t)=__ (Use integers or decimals for any numbers in the expression.) Answer: Step 1 Given that a stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3.4 ft/sec. Step 2 Since, in one second the increment in radius is 3.4 ft. So, after t second the increment in radius is given by r(t)=3.4t Step 3 Answer: The radius function is given by r(t) = 3.4t. Question #2 Solve the exponential equation. Express irrational solutions as decimals correct to the nearest thousandth. 5𝑥−3 = 32𝑥 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is ______. B. The solution is the empty set. Answer: Step 1 The given exponential equation is 5𝑥−3 = 32𝑥 . Step 2 5𝑥−3 = 32𝑥 ln(5𝑥−3 ) = ln(32𝑥 ) [If 𝑓(𝑥) = 𝑔(𝑥), then ln 𝑓(𝑥) = ln 𝑔(𝑥) ] (𝑥 − 3) ln(5) = 2𝑥 ln(3) [∵ log 𝑎 (𝑥 𝑏 ) = 𝑏 ⋅ log 𝑎 (𝑥)] ln(5) 𝑥 − 3 ln(5) = 2𝑥 ln(3) (ln(5) − 2 ln(3))𝑥 = 3 ln(5) 𝑥= 3 ln(5) ln(5)−2 ln(3) x=-8.21439 x=-8.214 [Rounded to nearest thousandth] Thus, the solution set is -8.214 Question #3 One way to find the formula of a line passing through (15,16) and (30,25) is by using a table, as shown below. Complete parts (a) and (b) below. a) Complete the box in the first row. x y 0 15 16 30 25 b) Report the formula of the line in slope-intercept form. y=___ x+ _____ Answer: Step 1 line passing through point (15,16) and (30,25) The equation of line is given by 𝑥 − 15 𝑦 − 16 = 30 − 15 25 − 16 𝑥−15 𝑦−16 = 15 9 5 𝑥 − 15 = 𝑦 − 16 3 3𝑥 − 45 = 5𝑦 − 80 3𝑥 + 35 = 5𝑦 3 𝑦= 𝑥+7 5 y=0.6x+7 Step 2 a) Complete box. x y 0 7 15 16 30 25 b) Report line in slope-intercept form y=0.6x+7. Question #4 Solve the exponential equation. Express irrational solutions as decimals correct to the nearest thousandth. 4𝑥−1 = 32𝑥 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is ______. B. The solution is the empty set. Answer: Step 1 x=1.724 Option a is correct Step 2 4𝑥−1 = 32𝑥 = 𝑥 − 1 = log 4 32𝑥 = 𝑥 − 1 = 2𝑥 log 4 3 𝑥−1 = = log 4 3 1 2 2𝑥 1 − 1 = log 4 3 2𝑥 = − log 4 3 = = 2 1−2 log4 3 2 =𝑥= =𝑥= =𝑥= =𝑥= = 1 1 2𝑥 1 2𝑥 1−2 log4 3 1 1−2(0.79) 1 1−1.58 −1 0.58 = 1.72 = 𝑥 = 1.724 Question #5 The results of a survey of 39 students and the foreign language they are studying are shown in the twoway frequency table. Gender Chinese French Spanish Total Girl 3 8 13 24 Boy 2 2 11 15 Total 5 10 24 39 What fraction of girls are studying French? Use decimals to answer, and round to 3 places. The fraction of girls that are studying French is ______. Answer: Step 1 Using table Given, number of girls studying French =8 And total number of girls =24 Step 2 So fraction of girls that are studying french number of girls studing french = total number of girls 8 = 24 1 = 3 =0.333 Hence, answer =0.333 Question #6 Write an algebraic equation for the following problem and then solve it. Beth and Mark borrowed $4000 at a simple interest rate of 5% for a period of 3 years. What was the interest? Write an equation for the interest I earned. I=__ (Do not simplify. Use integers or decimals for any numbers in the expression.) The interest is $___. (Simplify your answer.) Answer: Step 1 The equation for the interest earned is 𝐼 =𝑃 ⋅𝑟 ⋅ where I: Interest P: Principal r: Rate of interest t: Time in years Step 2 Substitute the given alues in the above expression to find the interest earned. P = $4000 r = 5% t = 3 years 𝐼 =𝑃 ⋅𝑟 ⋅ = 4000 × 5 100 ×3 =600 Step 3 Answer: The algebraic expression that gives the interest, I = Prt The interest obtained after 3 years = $600. Question #7 A stone is thrown into a pong, cheating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 2.4 ft/sec. Complete parts a through c. a) Find a function for the radius in terms of t. r(t)=__ (Use integers or decimals for any numbers in the expression.) Answer: Let r be the radius of the circular ripple and the radius is increasing at a rate 2.4 feet/second. That is, the rate of change is 𝑑𝑟 𝑑𝑡 𝑑𝑟 𝑑𝑡 − 2.4. = 2.4 𝑑𝑟 = (2.4)𝑑𝑡 Integrate on both sides as follows. ∫ 𝑑𝑟 = ∫(2.4)𝑑𝑡 ∫ 𝑑𝑟 = 2.4 ∫ 𝑑𝑡 𝑟 = 2.4𝑡 + 𝐶 24 𝑟= 𝑡+𝐶 10 12 𝑟 = 𝑡+𝐶 5 Thus, the function for the radius in terms of t is 𝑟(𝑡) = Question #8 12 5 𝑡 + 𝐶. The pmf of the amount of memory X (GB) in a purchased flash drive is given as the following. x 1 2 4 8 16 p ( x) 0.05 0.10 0.30 0.45 0.10 a) Compute E(X). (Enter your answer to two decimal places.) GB b) Compute V(X) directly from the definition. (Enter your answer to four decimal places.) 𝐺𝐵2 c) Compute the standard deviation of X. (Round your answer to three decimal places.) GB d) Compute V(X) using the shortcut formula. (Enter your answer to four decimal places.) 𝐺𝐵2 Answer: Step 1 The expected value of X is obtained below: From the given information, the random variable X represents the amount of memory in a purchased flash drive and it takes the values 1, 2, 4, 8 and 16. Consider, 𝐸(𝑋) = ∑𝑥=1,2,4,8,16 𝑥𝑃𝑋 (𝑥) = (1 × 𝑃((𝑥 = 1)) + (2 × 𝑃(𝑥 = 2)) + (4 × 𝑃(𝑥 = 4)) + (8 × 𝑃(𝑥 = 8)) + (16 × 𝑃(𝑥 = 16)) = (1 × 0.05) + (2 × 0.10) + (4 × 0.3) + (8 × 0.45) + (16 × 0.10) = 0.05 + 0.2 + 1.2 + 3.6 + 1.6 =6.65 The expected value of random variable X is obtained by summation of product of values of x and the corresponding probabilities. Step 2 The variance of random variable X is, 𝑉(𝑋) = 𝐸(𝑋 2 ) − [𝐸(𝑋)]2 Consider, 𝐸(𝑋 2 ) = ∑ 𝑥 2 𝑃𝑋(𝑥) (12  P( x = 1)) + (22  P( x = 2)) + (42  P( x = 4)) +  =  (82  P( x = 8)) + (162  P( x = 16))   = (12 × 0.05) + (22 × 0.10) + (42 × 0.3) + (82 × 0.45) + (162 × 0.10) = 0.05 + 0.4 + 4.8 + 28.8 + 25.6 =59.65 Thus, the value of 𝐸(𝑋 2 ) is 59.65 𝑉(𝑋) = 𝐸(𝑋 2 ) − [𝐸(𝑋)]2 = 59.65 − (6.65)2 =15.4275 The variance of the random variable X is obtained by subtracting the square of expected value of X from the expected value of 𝑋 2 The variance represents the square of the spread of the amount of memory in a purchased flash drive around the mean. Step 3 The standard deviation of random variable is, 𝜎 = √𝑉(𝑋) = √15.4275 =3.9278 The standard deviation of the random variable X is obtained by taking square root of the variance. The spread of the amount of memory in a purchased flash drive is 3.9278 times around the mean. Step 4 The variance of random variable X is, 2 𝑉(𝑋) = 𝐸 [(𝑋 − 𝐸(𝑋)) ] 2 = ∑𝑥 [(𝑋 − 𝐸(𝑋)) × 𝑃(𝑋 = 𝑥)] = ∑[(𝑋 − 6.65)2 × 𝑃(𝑋 = 𝑥)] 𝑥 [(1 − 6.65) 2  P( X = 1)] + [(2 − 6.65) 2  P( X = 2)] { +[(4 − 6.65) 2  P( X = 4)] } 2 2 +[(8 − 6.65)  P( X = 8)] + [(16 − 6.65)  P( X = 16)] [(1 − 6.65)2  0.05] + [(2 − 6.65) 2  0.1] + [(4 − 6.65) 2  0.3] { } +[(8 − 6.65)2  0.45] + [(16 − 6.65)2  0.10] = 1.5961 + 2.1623 + 2.1068 + 0.8203 + 8.7423 =15.4278 The variance of the random variable X is obtained by the expected value of square of difference between the random variable X and expected value of X. The square of the spread of the amount of memory in a purchased flash drive is 15.4275 times around the mean. Question #9 Determine 𝑧𝛼 for the following. (Round your answer to two decimal place) a) 𝛼 = 0.0089 b) 𝛼 = 0.09 c) 𝛼 = 0.707 Answer: Step 1 To find the value of 𝑍𝛼 at the significance level 0.0089, the confidence level would be: 1 − 0.0089 = 0.9911 Then use Excel function normsinv(0.9911) to get the corresponding z- value. The Excel formula used is as shown: A 1 = NORMSINV (0.9911) 2 Part a The Z-value for a significance level of 0.0089 is 2.37 At the significance level of 𝛼 = 0.0089 , the corresponding Z value is 2.37. Step 2 To find the value of 𝑍𝛼 at the significance level 0.09, the confidence level would be; 1-0.09=0.91 Use the Excel function normsinv(0.91) to get the Z value as shown: A 1 = NORMSINV (0.9911) 2 3 = NORMSINV (0.91) Part b The Z-value for a significance level of 0.91 is 1.34. At the significance level of \alpha = 0.09 , the corresponding Z-value is 1.34. Step 3 To find the value of 𝑍𝛼 at the significance level 0.707, the confidence level would be: 1-0.707=0.293 Use the Excel function normsinv(0.293) to get the Z value as shown: A 1 = NORMSINV (0.9911) 2 3 = NORMSINV (0.91) Part c 4 5 = NORMSINV (0.293) 6 The Z-value for a significance level of 0.293 is - 0.545 At the significance level of 𝛼 = 0.707, the corresponding Z value is - 0.545. Question #10 A sample of tritium-3 decayed to 94.5% of its original amount after a year. What is the half-life of tritium3? Answer: 𝐴 = 𝐴0 𝑒 −𝑘𝑡 t is in years After 1 year t=1 and 𝐴 = 0.945𝐴0 Substitute to get 1) 0.945 = 𝑒 −𝑘 Let half life be x. Then 𝐴 = 0.5𝐴0 Substitute to get 0.5 = 𝑒 −𝑘𝑥 0.5 = (𝑒 −𝑘 ) 𝑥 Substitute 0.945 − 𝑒 −𝑘 Using eqn(1) 0.5 = (0.945) 𝑥 Take 𝑙𝑛 on both sides to get ln(0.5) = 𝑥 ln(0.945) 𝑥= ln(0.5) ln(0.945) = 12.25 Half life is 12.25 years Question #11 Convert 3 8 to a decimal. Answer: We can do like this: 1 If = 0.5 1 4 1 8 2 = 0.25 = 0.125 And we have 1 3 8 , so 3 × = 3 × 0.125 = 0.375 8 Question #12 Write the expression in rectangular form, x+yi, and in exponential form 𝑟𝑒 𝑖𝜃 . 4 (√13 − 𝑖) The rectangular form of the given expression is ?, and the exponential form of the given expression is ? (Simplify your answers. Use integers or decimals for any numbers in the expressions. Round the final answer to three decimal places as needed intermediate values to four decimal places as needed.) Answer: Step 1 Given query is to find a the rectangular form and the exponential form of the expression. Step 2 4 Given: (√13 − 𝑖) : we may write it as 2 2 (√13 − 𝑖) (√13 − 𝑖) Now (𝐴 − 𝐵)2 = 𝐴2 + 𝐵2 − 2𝐴𝐵 (13 + 𝑖 2 − 2√13𝑖)(13 + 𝑖 2 − 2√13𝑖) 𝑖 2 = −1 (12 − 2√13𝑖)(12 − 2√13𝑖) (144 + 4 × 13𝑖 −2 − 4√13𝑖) (144 − 52 − 4√13𝑖) = 92 − 4√13𝐼 → rectangular form Now 𝑟 = √(92)2 + (4√13) 2 𝑟 = √8464 + 208 ⇒ 𝑟√8672 𝑟 = 4√542 𝜃 = tan4 (− 4√13 𝜃 = tan−1 (− ) 92 √13 23 ) Thus = 𝑟𝑒 𝑖𝜃 ⇒ 4√542𝑒 √13 ) 23 𝑖 tan−1 (− Question #13 At a certain school, the number of student tickets sold for a home football game can be modeled by 𝑆(𝑝) = 59𝑝 + 8100, where p is the winning percent of the home team. The number of nonstudent tickets sold for these home games is given by 𝑁(𝑝) = 0.2𝑝2 + 12𝑝 + 4100. a. Write an equation T(p) for the total number of tickets sold for a home football game at this school as a function of the winning percent p. b. What is the domain for the function in part a. in this context? c. Assuming that the football stadium is filled to capacity when the team wins 90% of its home games, what is the capacity of the school's stadium? Answer: Total of s(p) and n(p) gives the expression for T(p) Solution: S(p)=59p+8100 𝑁(𝑝) = 0.2𝑝2 + 12𝑝 + 4100 a) Total number of ticket sold =s(p)+n(p) = 59𝑝 + 8100 + 0.2𝑝2 + 12𝑝 + 4100 = 0.2𝑝2 + 71𝑝 + 12200 b) As given for pass percent p 0 ≤ 𝑝 ≤ 100 c) p=90% p=90 it's full capacity 𝑇(𝑝) = 0.2(90)2 + 71(90) + 12200 = 20210 Capacity of stadium 20210 Question #14 Write an equation in y=mx+b form for the linear relationship shown in the table. Wind-chill effect of 20-mile-per-hour wind Actual Temperature(t) Wind-Chill Temperature (w) 45 F 25 F 25 −3 F 5 F −31 F Answer: For First Question. t = actual temp. degree Fahrenheit w(t) = Wind chill temp. degree Fahrenheit. So, the point is (t, w(t)) Or, (45,25) ,(25,-3) and (5,-31) all values in degree Fahrenheit. So, Equation of line pass through from (45 ,25) and (25 ,-3) is Or, 𝑦 − (−3) = ((−3 − 25)/(25 − 45))(𝑥 − 25) Or, 𝑦 + 3 = (−28/−20)(𝑥 − 25) Or, 𝑦 + 3 = (14/10)(𝑥 − 25) Or, 𝑦 + 3 = 1.4𝑥 − 35 Or, 𝑦 = 1.4𝑥 − 35 − 3 Or, 𝑦 = 1.4𝑥 − 38 . Now , Check this equation for (5,-31) ,Put the value of x = 5 So, 𝑦 = (1.4)(5) − 38 Or, 𝑦 = 7 − 38 Or, y = - 31 Hence , this linear equation satisfied the all three points. So , The final answer is: y=1.4x-38 Question #15 Wind (mph) Wind-Chill( F ) 5 25 10 21 15 19 20 17 25 16 30 15 35 14 40 13 45 12 50 12 55 11 60 10 The table gives the wind chill temperature when the outside temperature is 30∘ 𝐹. a. Use x as the wind speed and create a quadratic model for the data. b. At what wind speed does the model estimate that the wind chill will be 14∘ 𝐹 ? c. Is the model found in part (a) valid for x>63? Explain Answer: Determining quadratic model for the data . Below is the step-by-step procedure. a) Let the wind speed be x. The wind chill can be expressed as 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 from given data y(5)=25 ⇒ 25 = 𝑎52 + 𝑏5 + 𝑐 25=25a+5b+c (1) y(10)=21 ⇒ 21 = 𝑎102 + 𝑏10 + 𝑐 21=100a+10b+c (2) y(15)=19 ⇒ 19 = 𝑎192 + 19𝑏 + 𝑐 19=361a+19b+c (3) Solving 1, 2 and 3 we get a=13/315=0.04/3 b=-149/105=-1.4190 c=1957/63=31.0635 wind chill 𝑦(𝑥) = 0.0413𝑥 2 − 1.419𝑜𝑥 + 31.0635 b) At x=?, y(x)=14 ⇒ 14 = 0.0413𝑥 2 − 1.4190𝑥 + 32.0635 ⇒ 𝑥 ≈ 34.86 . estimate of using speed is 34.86 mph. c) for x>63 . The model will be valid because it gives a general expression to determine f(x) any x. Question #16 A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. A graphing calculator displays a quadratic function that models the ball's height, y, in feet, in terms of its horizontal distance, x, in feet. Answer x, Balls Horizontal Distance(feet) 0 1 3 4 y, Balls Height(feet) 6 8.1 6 2.9 a. Explain why a quadratic function was used to model the data. In the quadratic regression screen shown in the problem statement, why is the value of the coefficient a negative? b. Use the graphing calculator screen (shown in the box above) to express the model in function notation. f(x)=? c. Use the model from part (b) to determine the k-coordinate of the quadratic function's vertex. The x-coordinate of the vertex is ? Answer: a) As we can see from the above table, that height of the ball first increases and then decreases, which looks like a quadratic function. The coefficient a in the regression screen is negative because the height of the ball increases and then decreases, therefore, the quadratic must open downward. b) Using the quadratic regression model, 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 a=-0.9, b=2.6, c=6.1 Thus, 𝑦 = 𝑓(𝑥) = −0.9𝑥 2 + 2.6𝑥 + 6.1 c) We know that, the vertex of the parabola 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 is given by 𝑥=− 𝑏 2𝑎 ⇒𝑥=− 2.6 2(−0.9) = 1.44 The x coordinate of the vertex is 1.4 The maximum height occurs at the vertex. Thus, 𝑦𝑚𝑎𝑥 = 𝑓(1.44) = −0.9(1.44)2 + 2.6(1.44) + 6.1 = 7.978 ≈ 8 feet The maximum height of the ball occurs 1.4 feet from where it was thrown and the maximum height is 8 feet. Question #17 Compute the product AB by the definition of the product of matrices, where 𝐴𝑏1 and 𝐴𝑏2 are computed separately, and by the row-comumn rule for computing AB.  −2 3  A =  3 5 ],  6 −2  5 −1 B=   −1 4  Set up the product 𝐴𝑏1 , where 𝑏1 is the first column of B. 𝐴𝑏1 =? ? where b1 is the first column of B. Calculate 𝐴𝑏1 where 𝑏1 is the first column of of B. 𝐴𝑏1 =? Set up the product 𝐴𝑏2 where 𝑏2 is the second column of B 𝐴𝑏2 =? Calculate 𝐴𝑏2 where 𝑏2 is the second column of B. 𝐴𝑏2 =? Determine the numerical expression for the first entry in the first column of AB using the row-column rule. Answer: We will use the product method of matrix to find solution to the problem. Set up the product 𝐴𝑏1 , where b, is the first column of B. 𝐴𝑏1 = −2,5 + 3, −1 =-10+(-3) =-10-3 =-13 Calculate 𝐴𝑏1 where 𝑏1 is the first column of B Ab_{1}=-13 Set up the product 𝐴𝑏2 , where 𝑏1 is the second column of B. 𝐴𝑏2 = −2, −1 + 3,4 ⇒ 2 + 12 ⇒ 14 Calculate 𝐴𝑏2 , where 𝑏2 is the second column of B. 𝐴𝑏2 = 14 For the first entry, ((-2)(5)+(3)(-1)) is correct Product AB=  −13 14   10 17     32 −14 Question #18 The exponential models describe the population of the indicated country. A, in millions, t years after 2010. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year? Country B: 𝐴 = 1173.3𝑒 0.008𝑡 Country C: 𝐴 = 33.5𝑒 0.017𝑡 Country D: 𝐴 = 121.7𝑒 −0.004𝑡 Country E: 𝐴 = 145.7𝑒 0.002𝑡 Answer: As we know that the exponential model that is decreasing is written as 𝐴(𝑡) = 𝐴0 𝑒 −𝑘𝑡 where, 𝐴0 is the initial population k is the proportionality constant t is time after 2010. From the information we see that Country D and E is of the model 𝐴0 𝑒 −𝑘𝑡 . Hence the model for country D and E is the population is decreasing. the population of country D is decreasing by (1 − 𝑒 −0.004 ) × 100 = 0.4% and the population of country E is decreasing by (1 − 𝑒 −0.002 ) × 100 = 0.2% Question #19 The chart to the right shows a country's annual egg production. Model the data in the chart with a linear function, using the points (0,52.7) and (4,61.4). Let x=0 represent 1994, x=1 represent 1995, and so on, and let y represent the egg production (in billions). Predict egg production in 2000. How does the result compare to the actual data given in hte table, 70.7? Year Egg production (in billions) 1994 52.7 1995 53.6 1996 55.4 1997 58.3 1998 61.4 1999 65 2000 70.7 Answer: Step 1 Let x represent the year and y represent the egg production . Here x = 0 represent 1994, x = 1 represent 1995 and so on. Step 2 We need to find linear model for the data using point (0,52.7) and (4,61.4) We know that the slope of line using two points is given by 𝑦 −𝑦 61.4−52.7 8.7 slope = 2 1 = = = 2.175 𝑥2 −𝑥1 4−0 4 Now equation of the linear model is given by 𝑦 − 52.7 = slope(𝑥 − 0) 𝑦 = 2.175𝑥 + 52.7 Egg production in year 2000 (x=6) 𝑦 = 2.175(6) + 52.7 = 65.75 billion Compare the predicted egg production ( 65.75 billion ) in year 2000 to the actual data (70.7 billion) Take the difference from the actual data =70.7-65.75 = 4.95 billion The result is 4.95 billion eggs less than the actual figure . Question #20 Year 1970 1980 1990 2000 2010 Hours 2020 1880 1730 1690 1570 The table to the right lists the annual hours worked by the average worker in a country for selected years. a) Let x represent the number of years after 1970. Find a formula in slope-intercept form for a linear function f that models the data. b) Interpret the slope of graph of 𝑦 = 𝑓(𝑥). c) Estimate the annual hours worked in 2018. Answer: SInce 'x' is number of years after 1970 Thus 1970 is at 𝑥1 = 0, 𝑦1 = 2020 1980 → 𝑥2 = 10, 𝑦2 = 1880 Now slopf will be 𝑦 −𝑦 1880−2020 𝑚= 2 1⇒𝑚= 𝑚= 𝑥2 −𝑥1 −140 10 10−0 ⇒ 𝑚 = −14 a) Equation will by 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 𝑦 − 2020 = −14(𝑥 − 0) 𝑦 − 2020 = −14𝑥 ⇒ 𝑦 = −14𝑥 + 2020 b) Slope is -14 Hours worked decreased, on average 14 houes I year c) For 2018; 𝑥 = 2018 − 1970 ⇒ 48 𝑦 = −14(48) + 2020 𝑦 = 672 + 2020 y=1348 hour