Answers Investigation Applications. a. 7 gallons are being pumped out each hour; students may make a table and notice the constant rate of change, which is - 7, or they may recognize that - 7 is the coefficient of t in a linear relationship between w and t. b.,9 gallons. This can be found using a table and finding the value when t =, or by substituting for t into the equation and finding w. c. hours. Students can find in their table as the corresponding t-value when w is,, or they can solve the equation w = - 7t +,9 for t when w =,. d. 7 hours. Students can find 7 in their table as the corresponding t-value when w is, or they can solve the equation w = - 7t +,9 for t when w =. e. w = - 7(t - 7). The original equation tells you that before the pump started working, there were,9 gallons of water in the pool, and that for every hour the pump emptied 7 gallons of water from the pool. In this equation, when t = 7, the amount of water is, or the tank is empty. f. The relationship is linear because there is a constant rate of change.. a. gallons. Students may use a table and notice the constant rate of change is -, after multiplying - 7 by, or they may recognize that - is the coefficient of t in a linear relationship between w and t. w = - 7(t - 7) w = - t +,9 b.,9 gallons. This can be found using a table and finding the value when t =, or by substituting for t into the equation and finding w. c..8 hours or hour, minutes. Students can find.8 in their table as the corresponding t-value when w is,, or they can solve the equation w = - 7(t - 7) for t when w =, after applying the Distributive Property:, = - t +,9, -,9 = - t -9 - = -t -.8 t The pump has been running for about.8 hours. d.. hours; students can find in their table the corresponding t-value when w is, or they can solve the equation w = - 7(t - 7) for t when w =. = - t +,9 -,9 = - t -,9 - = -t -. = t The pool will be empty in. hours. e. w = - t +,9. This equation tells you that before the pump started working there were,9 gallons of water in the pool, and that for every hour the pump emptied gallons of water from the pool.. a. gallons. Let m =, since miles have been driven after the last fill-up. From the equation, g =, meaning the tank holds gallons of gas. b. Approimately.7 gallons. g = -.7 gallons. c. Substitute for g into the equation, = m. Solve for m, so m = miles. Note: A graph and a table would also show that gallons remain after miles. Say It With Symbols Investigation
Answers Investigation d. 7 miles. Students may use their table to find the value of m that corresponds to g =, or solve the equation g = - m for m when g equals. Since m has a coefficient of -, students may have a difficult time deciding how to apply the properties of equality, they may multiply by, or they could also divide by. e. The tank holds gallons, so g = - = when gallon has been used. Therefore, = - m, so m =. The driver would have to travel miles to use gallon of gas. f. is the number of gallons of gas in the tank after a fill-up, and indicates that the truck uses of a gallon of gas every mile.. The variable y represents how much money they still need to pay for the printing bill, depending on the number of books sold. N represents the number of books sold or given away,, is the amount they owe for printing at the start of the project, is the price they charge for each book, and 8 represents the free copies they gave to the yearbook advisor and staff.. a. A = /( - /); since A = /w, you need to write w in terms of /. Perimeter is from the given amount of fencing. So, using the equation = / + w and solving for w, w = - / and A = /( - /). b. The maimum area is when / and w each equal. c. If you graph the equation A = /( - /), you get a parabola that opens down. To find the maimum area, you look at the maimum point on the parabola, or the verte. The -coordinate is the length of the rectangle with the greatest area. To find the width, substitute this -coordinate into the equation w = - / and solve for w. d. The equation is quadratic because it is the product of two linear factors that are in terms of /. The equation A = /( - /) can be written as A = / - /, where the eponent on / is. Also, is the highest eponent to which / is raised.. a. A = /( -./). Since A = /w, write w in terms of /. Since the fencing is meters, use the equation = w + / + w. Solve to find w = -./. b. The length would be meters, so the width could be found using the equation w = -./. The width would be meters. c. The equation is quadratic because it is the product of two linear factors that are in terms of /. The equation A = /( -./) can be written as A = / -./, where the eponent on / is. This is the highest eponent to which / is raised. 7. a. First you need to write the equation n = - in terms of = - n. So, substituting into P = n - n, P = ( - n)n - n, or P = n - n - n, which is equivalent to P = n - n. b. +; using the equation P = n - n, substitute for n. The profit is P = () - =. c. The selling price can be found using the equation n = -. So, when n =, the selling price is +. d. +9; the greatest profit can be found by making a table or graph for the profit equation P = n - n - n = n - n. The greatest profit occurs when 7 posters are sold, which yields a value of P = (7) - 7 = 9. Say It With Symbols Investigation
Answers Investigation 8. a. Table is quadratic with a second difference of. Table is linear with a constant rate of change of. Table is eponential with a growth factor of. b. Possible answers: Table : Let N be the number of deer and be the number of years after (so when =, the year is ); then the equation is N = + 9 +,. Table : Let N be the number of deer and y be the year. Then N =, + (y -,). Or, let N be the number of deer and be the number of years after (so when =, the year is ); then the equation is N =, +. Table : Let N be the number of deer and y be the year. Then N =,() y-,. Or, let N be the number of deer and be the number of years after (so = represents the year ), then the equation is N =,(). c. Table shows the deer population growing at a rate of, per year. 9. a. For Species,, is the starting population. is the rate at which the population grows every year. So, for every year the population increases by animals, P is the total population after years. For Species, is the starting population, and t means that the population triples every year. P is the total population after t years. For Species, 8 is the starting population and t means that the population increases by the product of and the number of years passed multiplied by itself. P is the total population after t years. b. The pattern of growth for Species is linear. The pattern of growth for Species is eponential. The pattern of growth for Species is quadratic. After a certain time, the population of Species will surpass the other two populations, since eponential growth patterns increase at an increasing rate. Say It With Symbols c. Answers will vary; however, any two populations will be the same at some value for t. The populations of Species and are the same when t is between. and.. The populations of Species and are the same when t is between. and.. The populations of Species and are the same when t is about.7 and -.7. A negative answer does not make sense in this situation, so -.7 is not a solution. One way to find these values for t is to use a graphing calculator. If you use the table function and set the increments to. or., you can get close estimates for the values for which the equations are equal.. a. Answers will vary. b. 7. Students may draw the net two figures and count the number of toothpicks, or make a table of values and use the pattern in the table to find the number of toothpicks in the seventh figure. Figure 7 Toothpicks 8 8 7 c. Linear. Possible answer: The figure number times equals the perimeter. The figure number equals the number of toothpicks on the bottom and the number of toothpicks going up (height). If you double the figure number, you get the number of toothpicks that make up the stairs on the left side of the figure giving n + n + n = n. This pattern shows that the data will go up at a constant rate. The graph will be a straight line with a slope of. Investigation
Answers Investigation d. Quadratic. Possible answer: In the data table, as increases by, the y-value has a second difference of. e. P = N. To find the perimeter, you take the figure number and multiply it by. f. Possible answers: T = N + N, where T is the total number of toothpicks and N is the figure number. If you work backward on the table, you find that the y-intercept is. This means that in the quadratic equation form of y = a + b + c, c =. Because the second difference is, the value of a =, since a = the second difference. Substituting into the quadratic equation, y = + b + = + b. You can use a table to find b. See below. Figure Number 9 b + ( ) + ( ) + ( ) + 9( ) + ( ) Total Toothpicks 8 8 + T = N(N + ), where T is total number of toothpicks and N is figure number. Figure (N) N + 7 Total Toothpicks N(N + ) 8 8 If you divide the total number of toothpicks by the figure number, the result is the second column of numbers. This number is the figure number plus. To get the total number of toothpicks, you multiply the N + and the figure number.. Graph is linear since it is a straight line with a constant rate of change of. Graph is eponential since it has an increasing graph with a growth factor of. Graph is quadratic since it has an upside-down U shape and a second difference of -. Graph is an inverse variation and was studied in Thinking With Mathematical Models.. Graph y 7 Graph y 8 Graph y 9 7 8 79 Graph y.....7. Graph : y = -. To find this equation you need to find the y-intercept and the slope or rate of change. Students may use the formula m = y (i.e., m = y - y - ) to find slope by using two of their points in the table. They may look at the constant rate of change, which is. To find the y-intercept, they may look at the graph and see that it is -. Say It With Symbols Investigation
Answers Investigation. G. D. B 7. F 8. E 9. C. A Graph : y =. To find this equation, students need the starting point and the growth factor. By looking at the table, each y-value increases by a growth factor of. The starting point can be found by dividing the y-value for = by to get the y-value for =. Doing this, y = for the starting value. So, the equation is y = () or y =. Graph : Since the -intercepts are and, the factors could be ( - ), and the equation could be y = ( - ). By checking the point (, ) in this equation, you can verify that this is correct, since three points, the -intercepts and the point (, ), determine a parabola. Graph : Graph is neither linear, eponential, nor quadratic. It is an inverse variation that students studied in Thinking With Mathematical Models. Note: The equation for Graph is y =, or equivalently y =.. a. Linear: Equations and. Quadratic: Equations,, 8, and 9. Eponential: Equations,, and 7. b. Equations and represent the same function. Equations and represent the same function. Equations and represent the same function. c. The graph of Equations and is a line with a starting point of (, ), a rate of change of, and an increasing pattern from left to right. The graph of Equations and is a parabola that opens up with a y-intercept of (, ), -intercepts of (, ) and ( - 8, ), and a minimum point at ( -, - ). The graph of Equations and is an increasing curve with y-intercept (, ) and a growth factor of. Say It With Symbols. Answers will vary. Possible answer for the linear equation is y =. You get dollars for every kilometer you walk, where is the number of kilometers walked and y is the total amount of money collected. Possible answer for the quadratic equation: y = + 8. This represents the number of handshakes between two teams if one team has members and the other team has + 8 members. Possible answer for the eponential equation: y = -. y is the number of rubas on the th square of a checkerboard if the King puts on the first square, on the second, and on the third and then continues to quadruple the number of rubas for each successive square.. a. Linear: Equations, 7,, 8 Eponential: Equations,, Quadratic: Equations, 9,,,,, 7 b. Equations and 9; Equations and ; Equations and ; Equations and 7 c. Equations and 9: Quadratic pattern with y-intercept of and -intercept of -. The minimum is ( -, ). Equations and : Eponential patterns with starting point (, ) and a growth factor of. Equations and : Linear pattern, -intercept is (, ) and y-intercept is. The rate of change is -. The line has a negative slope, so it falls left to right. Equations and 7: Quadratic pattern with y-intercept of (, ) and -intercepts of (, ) and (, ). The maimum is (., ).. Answers will vary. An eample for equation is that the King of Montarek will put one ruba on the first square on a chessboard, on the net square, 9 on the net square, and so on, multiplying by for each successive square. Equation represents the number of rubas on square of the chessboard. Investigation
Answers Investigation Connections. a. Plan : 7 weeks; Plan : weeks; Plan : 8 weeks Cookie Prices Week 7 8 9 Price (dollars) Plan Plan Plan...............7.7.8......8...8...7.8........7.8 b. y 7 Plan Plan Plan Week Price (dollars) c. The graphs of Plans and are linear because they are straight lines. The price of cookies in Plan grows at a constant rate of +. per week, and the price of cookies in Plan grows at a constant rate of +. per week. Plan is not linear because it grows by +. from week to week, and then the amount of growth increases each week. Plan is eponential growth. d. Possible answer: Customers will notice the change in Plan more quickly than in either of the other plans. Plan would be the least noticeable to customers. Note: Students should justify their choice based on the mathematics of the problem.. a.,, -, =, where is the number of days since the canister was opened;,,,, = days b. Epected Chip Count Number of Chips (thousands), 8 y 8 Day c. y =-, +,, Say It With Symbols Investigation
Answers Investigation Number of Chips (thousands) d., 8 Epected vs. Actual Chip Count y 8 Day Betty is going to suspect something strange is happening. The decrease in the number of chips is not constant and is greater than what is epected. 7. a. Cookie Price Survey Number of Customers y..... Price (dollars) b. Answers will depend on the curve that students draw. About customers would be willing to pay +.; about 9 would be willing to pay +.. c. Answers will vary. The predictions above are probably pretty accurate. d. Students may say that this graph model is similar to the inverse relationships they have seen in the bridge-length, teeter-totter, and travel-time problems from Thinking With Mathematical Models. They may also say it is like an eponential decay problem they have seen in Growing, Growing, Growing. The graphs of inverse relationships and decreasing eponential relationships have different shapes, but this is Say It With Symbols 7 probably too subtle a point for students who have only seen a few eamples of each at this point and time. 8. a. First, Tara distributed ( + ) to and. Second, she distributed to and. Third, she distributed to and. Then she applied the Distributive Property when she said that + = ( + ). To get the term, she used the Commutative Property: ( + ) = ( + ) =. b. Finding the area of the left-hand column of the table, which is ( + ), and adding it to the right hand column area ( + ), is the same as her first step. Her second step is just epressing the two column s areas as a sum of the parts that make them up. The last two steps are just to combine the two -terms and are not represented in the area model. c. i. + 8 + ( + )( + ) = ( + ) + ( + ) = + + + = + ( + ) + = + 8 + ii. + + ( + )( + ) = ( + ) + ( + ) = + + + = + ( + ) + = + + iii. + - 8 ( - )( + ) = ( - ) + ( - ) = - + - 8 = + ( - + ) - 8 = + - 8 9. a. ; ( * ), =, = n(n - ) b. h = c. h = n - n or h = (n - n). a. + 7 + or + + + b. Answers will vary. Students may use an area model to justify that their epressions are equivalent, or they may use a graph or a table to show that their epressions are equivalent. Note: If three points satisfy different quadratic epressions, then the epressions are equivalent. Investigation
Answers Investigation. a. y-intercept is ; students may find this by looking at a graph or a table when =...... 7. b. The -intercepts are - and -. Students can find the -intercepts by looking at a graph or a table when y =. c. The minimum is at = -. where the value of y is -.. The students may use a table or graph. There is no maimum. d. The line of symmetry is the vertical line through the value = -.. y 8 8 7 88 8. a. +,; she gets, *. =,. b. +,;, *. =,. c. +,. A, increase means that the new salary is, of the original salary. Thus, the salary is +, *. = +,. 9. a. Cylinder A is fatter and shorter than either of the other cylinders. Cylinder C is the same height as Cylinder B but skinnier. Cylinders B and C are both twice as tall as Cylinder A. b. Cylinder A has the greatest surface area. Students may count squares on the grid pattern to estimate the surface area. If they use formulas, they will probably use the actual measurements. Cylinder A has radius and height and surface area = p() + p()() = p. Cylinder B has radius and height 8 and surface area = p() + p()(8) = 8p. Cylinder C s surface area is less than Cylinder B s surface area since its rectangle and circles are smaller. c. Cylinder A; since volume equals pr h, find V when r h is the greatest. For Cylinder A, r h = ; for Cylinder B, r h = 8; and for Cylinder C, r h =.. a. 7.9 feet; d =-(. ) + (.) +. = 7.9 b. 9.8 feet; d =-(. ) + (.) +. = 9.8. Note: Ask students if this is reasonable. c.. feet; d =-( ) + () +. =. d. The operations are eponentiation, multiplication, and addition. The eponentiation is done first, then the multiplication, and lastly the addition. Note: The multiplication of numbers not involving eponents could be done before the eponentiation.. a., and,; to find the population after hours, substitute into the equation b =,( t ) for t. Then b =,(8) =,. To find the population after hours, substitute into the equation for t. Then b =,() =,. b. First, perform the repeated multiplication defined by the t in the parentheses; then take this product and multiply it by,. Say It With Symbols 8 Investigation
Answers Investigation. Possible answers: - 9 -, - -, or - - -. Possible answers: - + 9, - + - ( + ), - + - -, - ( - ) - -, - + -, or - + 9 -. Possible answers: + - +, or - +. Possible answers: + - +, or - +. Possible answers: + +, 7 +, ( 7 + ), or 7 ( + 7 ) 7. Possible answers: 7 -.7-8, or.( - ) +.( - ) 8. a. y = (z - 7) +, or equivalently, y = z - +, or y = z - b. P = ( - n)n - n, or equivalently, P = n - n - n, or P = n - n, or P = n( - n) c. A = ( - w)w, or equivalently, A = w - w 9. Possible answer: y = ( + )( - ) = + -. Some students may have equations that are quadratics in factored form of a( + )( - ), where a is a nonzero real number. As long as the linear factors have - and as their solutions for when the factor is set equal to zero, the answer is valid. Also, equations that are not of the form a( + )( - ) may work, too. For eample, ( + )( - ), which epands to y = + -, is a possible answer.. y = - + 8 is the only possible equation unless the student writes another equation that is equivalent to this.. Possible answers: y =., y = a(.), where a is a real number.. a. (See Figure.) Figure y = y 8 y = 8 8 8 y = y = Say It With Symbols 9 Investigation
Answers Investigation b. If a is positive, then the parabola opens up and if a is negative, then the parabola opens down. As a increases, the parabola becomes narrower, and as a decreases, the parabola becomes wider.. a. (See Figure.) b. The c-value is the y-intercept, so changes in the c-value move the parabola up or down. If c is, the y-intercept is at the origin, and when c increases, the parabola moves up, since the y-intercept value is increasing. As c decreases, the parabola moves down, since the y-intercept value is decreasing.. About.7 feet. To find h in the diagram above right using the Pythagorean Theorem, solve the equation h + ( h ) =, which is the same as solving the equation h + h =,, or h =,. Students may either divide each side by to obtain the equation h =,88, or they may look at the table of y = h on the graphing calculator to find h when y =,. h ft h ft ft. Cone ; the base area of the first cylinder is p. The volume of cylinder is p (), so the volume of the cone is a third of that, or p. The base area of the cylinder on the right is p() = 9p, so the volume of this cylinder is 9p () and the volume of cone is a third of the volume of the cylinder, or p cubic units. Figure y = + y y = + 8 O 8 y = y = Say It With Symbols Investigation
Answers Investigation Etensions Cost (dollars). a. Plan. If Caley uses Plan, she will owe +. * = +7. If she uses Plan, she will owe +. So, Plan is the better choice. 7 b. The equations are the same when you use minutes. In order to pay + for Plan, you would pay + for up to minutes and then + more for more minutes for a total of + = minutes. c. Suppose that M and M are the monthly bill amounts and n is the number of minutes used. Plan s equation is M = when n is from to minutes and M = + ( - n). for more than minutes. Plan s equation is M =. The growth pattern for M is linear and M is linear in pieces. y Cell Phone Offers Number of Minutes Plan II Plan I d. Plan is a horizontal line, and then after minutes, it has a positive slope. Plan remains a horizontal line no matter how many minutes you use. So, Plan will always cost more money if you talk more than minutes. (CR + YR + TR + IR) 7. Overall Rating = (.8 +.97 +.8 +.8). Since the Completion Rating is CR = ( 88 7) -..7, CR.8; the,9 7 Yards Rating is YR = -.7, YR.97; the Touchdown Rating is TR = ( 7) 8.7, TR.8; and the Interception Rating is IR = 9 - ( 7).7, IR.8. 8 Note: The four statistics, CR, YR, TR, and IR, cannot be negative or eceed.7. When a statistic is negative, then is used in the Overall Rating for that statistic. When a statistic eceeds.7, then.7 is used in the Overall Rating for that statistic. 8. a.. feet; a car should be S = + + 8, or. feet away. b. mph; ft/sec is * =, feet per minute and, * = 8, feet per hour. Then 8,, = miles per hour c. About 7. or 8 ft/s (.7 or.9 mi/h). If a car is trailing feet behind a car, the car s safe speed would be 7. or 8 feet per second, which is about.7 or.9 miles per hour. Students can find these values by putting the equation S = v + v + 8 into a graphing calculator and using the table to find the value for v when S is. 9. a. Solutions of ( + )( - )( - ) = are = -, =, and =. Those solutions are shown on the graph; they are the points where the graph crosses the -ais. b. The values of that satisfy ( + )( - )( - ) are - and. This can be seen on the graph where portions of the curve are below the -ais. Say It With Symbols Investigation
Answers Investigation c. Using only the equation and answers to part (a), you can find answers to part (b) by substituting a number less than for in the epression ( + )( - )( - ) and asking if the result is positive or negative. Repeat the process for a number between - and, for a number between and, and for a number greater than. When the result is a negative number for the chosen interval, the -values in that interval satisfy the given inequality ( + )( - )( - ).. a. Graph y = +. It is not possible to find when y is zero, since there are no -intercepts. Also, solving the equation = + means finding a number that, when you square it and add, gives you zero, which is not possible with real numbers because whether negative or positive, a number squared is positive. Furthermore, adding only results in a positive number. Therefore, the result cannot be zero. b. Answers will vary. Some possible answers: = +, = + + 8. Note: Any answer for y = a + b + c in which the value of b - ac is a negative number is a possible answer. This is because in the quadratic formula given in the Did You Know? after Problem., if b - ac is negative, the result is a negative value under the radical in the formula. This results in roots that are not in the real number system. When this happens, the parabola does not cross the -ais in the coordinate (real) plane. c. Answers will vary. Some possible answers: = + +, = + 8 + ; any quadratic that can be factored into the form a(y + z) where a, y, and z are real numbers. d. Answers will vary. Some possible answers: = -, = + + 8. Say It With Symbols Investigation