Chapter 32 Exercise 32.1

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1 Chapter Exercise. Q.. (i) x + y = x = y = y = x = y = x = (,) (,) x + y = (,) (,) 7 (ii) x + y = x = y = y = x = y = x = (,) (,) x + y = 7 (,) (,) Active Maths Strands Ch Solutions

2 (iii) 7x y = x = y = y = 7x = y = 7 x = (, 7) (,) 7x - y = (,) 7 7 (,7) (iv) x y = x = y = y = x = y = x = (, ) (,) (,) 7 9 (, ) Active Maths Strands Ch Solutions

3 (v) x y = x = x = y = () y = x y = (,) y = y = (,) y = y = (,) (,) (vi) x + y = x = y = y = x = y = x = (,) (,) (,) x + y = (,) Q.. (i) y = x + x = y = y = x = (,) x = (,) y = x + (,) (,) Active Maths Strands Ch Solutions

4 (ii) y = x + x = y = y = x = (,) (,) y = x + (,) (,) (iii) y = x + x = y = y = x = (,) x = y =.x + (,) (,) (,) (iv) y = x + x = y = y = x = (,) x = 9 (9,) y =.x + (,) (9,) 7 9 Active Maths Strands Ch Solutions

5 (v) y = x x = y = y = x = (, ) x = x = (,) y =.7x (,) 9 7 (, ) 7 (vi) y = x x = x = y = y = (,) (,) 7 (,) y = x (,) Active Maths Strands Ch Solutions

6 Q.. f: x x Q.. 7 f: x x 9 7 Q.. (i) (,) (ii) (,) (iii) (, ) (iv) (, ) Q.. (i) x + y = x = y = y = x = y = (,) (,) x y = x = y = y = x = y = (,) (, ) POI: (,) x + y = (,) x y = 7 9 (ii) x + y = x = y = y = x = (,) (,) POI : (,) x + y = x = y = y = x = (,) x = (,) Active Maths Strands Ch Solutions

7 x + y = x + y = 9 7 (,) 7 (iii) x y = x = y = y = x = y = (,) (,) POI: (,) x + y = x = y = y = x = y = x = (,) (,) 9 7 (,) Active Maths Strands Ch Solutions 7

8 (iv) x y = x = y = y = x = y = x = (,) (,) POI: (, ) x y = x = y = y = x = y = (,) (, ) x y = 7 x y = (, ) (v) x + y = x y = x = y = x = x = y = x = y = y = y = (,) y = y = (,) POI: (,) (,) (,) x + y = (,) 7 x y = Active Maths Strands Ch Solutions

9 (vi) y = x + x = y = y = x = (,) x = (,) POI: (7,) y = x x = y = y = x = (, ) x = (,) y = x y = x + (7,) (vii) y = x + y = x + x = y = x = y = y = x = y = x = (,) (,) POI: (,) (,) x = (,) y = x + (,) y = x + Active Maths Strands Ch Solutions 9

10 (viii) y = x + x = y = y = x = (,) (,) POI: (,) y = x + x = y = y = x = (,) x = (,) y = x + y =.x + (,) (ix) y = x x = y = y = x = (, ) x = (,) POI: (,) y = x x = x = y = y = (,) (,) y = x (,) 7 9 y =.7x Active Maths Strands Ch Solutions

11 (x) x y = x = x = y = () y = y = y = (,) y = y = POI: (,) (,) y = x + x = x = y = y = + (,) y = (,) y =.x + (,) 7 x y = Q. 7. (i) No. x y = x y = 7 9 Active Maths Strands Ch Solutions

12 (ii) Yes. x + y = x + y = 7 9 (iii) Yes. 7 7 y = x y = x 7 9 Active Maths Strands Ch Solutions

13 (iv) No. y = x + y = x Exercise. Q.. (i) (ii) (iii) km (iv) y =.x + (v) Cost =.() + = Q.. (i) (ii) 9 (iii) km (iv) y = x + (v) Cost = () + = Q.. (i) (ii) 7 (iii) 9 units (iv) Cost =.(,) +. =. Q.. (i) (ii) (iii) units (iv) y =.x + (v) Cost =.() + = Active Maths Strands Ch Solutions

14 Q.. (i) Fahrenheit 7 9 Celsius (ii) F (iii) C (iv) F Q f(x) = x + (iii) f(x) = x + g(x) = x 7 (i) (ii) days (iv) 7 Q. 7. (i) (ii) (iii) 7 people (iv) y =.x + Q. 9. (i), (ii) 9, (iii) (iv) Apartment:, (v) Cost =.() + = Car:, Q.. (i), Total:, (ii), Yes, he will have enough. (iii), (iv) y =,x +, (v) Value =,() +, =, Active Maths Strands Ch Solutions

15 Exercise. Q.. (i) Week Savings ( ) (ii) 7 (iii) Savings (EURO) Weeks Q.. (i) Cost ( ) Call-Out Change hr hrs hrs hrs 7 (ii) Cost (EURO) Time (hrs) (iii) No. The line does not pass through (,). (iv) (v) 9 Q.. (i) Month Comics (No.) Jan. Feb. 7 Mar. Apr. 7 May (ii) This is a linear pattern, since the first difference is constant. (iii) + () = 77 Q.. (i) Time (in Weeks) Height (in cm) 7 (ii) cm (iii) Linear, since the first difference is constant. (iv) + 9() = 9 cm (v) = = weeks Active Maths Strands Ch Solutions

16 Q.. (i) No. of kilometres run Week Week Week (ii) Linear. The first difference is a constant. (iii) Km Week Week Week Week Week Week Week 7 (iv) After the first three weeks of training, he may decide to increase the rate at which he trains. In this case, the second part of the line graph would be steeper than the first part (the first weeks). Q.. (i) Radiators (Nos) Cost ( ),,,,7,,,7 7,7 (ii) Cost (EURO) No. of Radiators Active Maths Strands Ch Solutions

17 (iii), + () =, +, =, (iv),, = Q. 7. (i) No. of Days Cost ( ), 7, 9,,,, 7, 7 9,, (ii) Cost (EURO) No. of days 7 9 (iii) After Day (iv), + (,) =, (v),,, = 7, profit Q.. (i) No. of Days Seán Ciara (ii) 7 No. of Words Ciara Seán No. of days 7 9 (iii) (iv) = 9 = days Active Maths Strands Ch Solutions 7

18 Q. 9. (i) No. of Weeks Quantity Supplied (litres),,,,, 7, 9,. Quantity Supplied (litres) No. of Weeks (ii),,, =, = weeks, Q.. Milk (litres) Cheese (kg),,,,,, Cheese (kg) Milk (litres) Active Maths Strands Ch Solutions

19 Q.. (i) No. of books Cost ( ),, 9, 7 9, 7,, Cost ( ),,,,,,, 9 7 Number of Books (ii) Mad Printers No. of books Cost ( ) Cost ( ),,,,,,, 9 7 Number of Books (iii) Cheap Print:,; Mad Printers: 7, (iv) The cost of printing is the same for either company. (v) Mad Printers Exercise. Q.. (i) Height (km) Time (minutes) 7 9 (ii) 9 km (iii). minutes = minute seconds Active Maths Strands Ch Solutions 9

20 Q.. (i) Q (ii) Exponential (iii) days:. days:. 7 days:.9 Ans:.9 km Area Covered (sq. km) Days 7 (ii) 7. m/s (iii) s (iv). and. seconds Q.. (i) 7, (ii) Quadratic (iii), (iv) am. This is the time after which the population declines. Q.. (i) Quadratic (as it declines to a minimum value and then rises) (ii) Month (iii) Month (iv) 7 km = % 7 = % 7 = % Answer km Q.. (v) Months 9, and (i) Weeds grow more vigorously during the summer. This is indicated during months 9 to in the graph. x y (ii) 9 m (iii) second and seconds after release (iv) seconds after release Active Maths Strands Ch Solutions

21 Q. 7. (i) Height (cm) D E F C B A f Day..... (ii) cm (iii) Day and (iv) Day and Day (v) 7 cm Q.. x y 7 (i) C 9 7 f(x) = x x (ii) x. From approximately : (iii) From x = to x =.. hours (iii) hour +. hours =. hours Exercise. Q.. (i) = km (ii) : (iii) Along the line segment [DE] Active Maths Strands Ch Solutions

22 Q.. Distance (km) Time (minutes) Q.. Left school for a class trip at 9. walked km to the local museum which we reached at :. Spent one hour there before leaving and travelling at a quicker pace to the bus stop which was km away. Got the bus at. and went on a coach tour over a scenic route ( km eventually from school) before returning to school at.. Q.. Fig A Statement Fig B Statement Fig C Statement Fig D Statement Fig E Statement Q.. (ii).7 secs. (iii) m/s (iv) 7 7 G F E D C B A Time (sec) Speed (m/sec) Active Maths Strands Ch Solutions

23 Q.. (i) Race A: Oisín; Race B: Juan (ii) Race A: m; Race B: m (iii) Race A: Oisín seconds; Juan seconds Race B: Oisín seconds; Juan seconds (iv) Oisín: = m/s Juan: = m/s (v) seconds: start of race seconds: point of intersection of two graphs (vi) Juan: He starts approximately m ahead of Oisín at seconds. (vii) Juan: The slope of his graph at the start is greater than the slope of Oisin s graph. This shows that his rate of change is greater. (viii) Oisín: The slope of his graph at the finish is greater than the slope of Juan s graph. Revision Exercises Q.. (i) x + y = x = y = y = x = (,) (,) POI: (,). x y = x = y = y = x = (,) (,) (,) Active Maths Strands Ch Solutions

24 (ii) x y = x = y = y = y = (, ) POI: (,) x = (,) x y = x = y = y = x = y = x = = (,) (,) x + y = (,) 7 x y = (iii) x + y = x = y = y = x = y = x = = 9 (,) (,) POI: (,) y = ( ) x + x = y = y = x = (,) x = (,) x + y = (,) y =.x + Active Maths Strands Ch Solutions

25 (iv) y = x x = x = y = y = (,) (,) POI: (,) y = x + x = x = y = y = + (,) y = (,) (,) y =.x + y = x Q.. (i) = (ii) They earn per hour. (iii) After hrs: (iv) y = x x = y = () Ans = (v) The equation of the line method is more accurate if the correct equation is used. Reading the graph, as drawn, will involve some error (most likely). Q.. (i) 7 Cost (EURO) (ii) Initial set-up costs. (iii) 9 (iv) 7 [y = () + = 7] Number of Magazines Active Maths Strands Ch Solutions

26 Q.. (i) Time Money owed st month nd month rd month 7 (ii) Money Owed (EURO) Months 7 9 (iii) More than months. (Between and 7 months) Q.. (i) Time (hours) Cost ( ) Q. 7. (ii) y = x + (iii) Call out charge. (iv) Additional cost per hour. (v) y = (.) + = (i). (ii) at a.m (iii) a.m and a.m Q.. (i) : (ii) Between : and 9: km (iii) =.9 km/hr.7 hrs Active Maths Strands Ch Solutions

27 Q.. (i) Yes y = x y = x (, ) (ii) No y =.x + y =.x (iii) Yes x + y = 9 x + y = 7 (.,.) Active Maths Strands Ch Solutions 7

28 (iv) No Reasons: (i) Slopes are different. (ii) Slopes are equal (i.e. parallel lines). (iii) Slopes are different. (iv) Slopes are equal (i.e. parallel lines). Q. 9. (i) About 9 m (ii) The train travels slowly for the first seconds. Its speed then increases from around seconds into the journey. The fastest speed is between and seconds. (iii) It is a train leaving a station. Q.. (a) John walks to school at a constant moderate speed. He looks at his watch, realises he will be late if he doesn t hurry up, so increases the pace at which he walks to school and walks the final part of the journey at this faster speed. (b) A car turning a sharp bend in the road slows down into the bend and speeds up out of it. (c) The rate at which water fills a storage tank is constantly increasing. (d) A pizza company charge a set delivery charge plus a rate of x per pizza bought. (e) A car is stuck in traffic and does not move. (f) A train travels at a constant speed before slowing down to a stop at a train station. Q.. (i).7 (ii). a.m (iii). p.m and. a.m. Q.. (i) g (iv) y = 7 x (ii) 7 ounces (v) y = 7 () =,. g (iii) 7 Q.. (i) Number of Birds Day (ii) Quadratic The second differences are constant. (iii) Days and (iv) Day : ( + ) Day 7: 9 ( + ) Active Maths Strands Ch Solutions

29 Q.. (i) Height (cm) (ii) It is constant. (iii). cm (iv).. cm Q.. (i) Days: Cost: Cost per Day = = (ii) (iii) Cheaper Vans: Linear pattern (with fixed fee): There is an initial fixed fee of followed by a charge of per day. Vans R Us: Linear pattern (proportional): There is no fixed fee and a cost of per day. (iv) Cheaper Vans (v) Vans R Us: Cost = Time Cost = =, Cheaper Vans: Cost = + Time Cost = + =, Q.. (i) Exponential (ii) Day No. of Words 7, 9,,9,9, No. of Words Known Days (iii) Day :, Active Maths Strands Ch Solutions 9

30 Q. 7. Q.. (i) Height (metres) Time (seconds) (ii) Quadratic (iii) Height =. m. (i). (ii). m (iv) The pigeon had landed and is now taking off again. (iii). m Q. 9. (i) No. of Cans Level 7 9 (ii) = 9 cans (iii) Seven levels in total 7 = cm Active Maths Strands Ch Solutions

b) According to the statistics above the graph, the slope is What are the units and meaning of this value?

b) According to the statistics above the graph, the slope is What are the units and meaning of this value? ! Name: Date: Hr: LINEAR MODELS Writing Motion Equations 1) Answer the following questions using the position vs. time graph of a runner in a race shown below. Be sure to show all work (formula, substitution,