FOUNDATIONS OF MATH 11 Ch. 8 Day 1: RATES RATIOS AND PROPORTIONS A ratio is a comparison between two amounts. example: A recipe uses nine cups of flour to make four cakes. In this cake recipe, the ratio of flour amount to number of cakes is 9: or 9 A proportion is a statement equating two ratios. example: A proportion using the ratio above is 9: = 18:8 or 9 = 18 8 RATES A rate is a comparison of two amounts that are measured in different units. example: In the cake recipe, the rate of flour amount in a number of cakes is 9 cups of flour 9 cups of flour per cakes or cakes A unit rate is a rate in which the numerical value of the second term is 1. example: In the cake recipe, the unit rate of flour amount in a cake is.5 cups of flour in a cake or.5 cups of flour/cake example: $50 can buy 36 litres of gasoline. What is the unit price? cost rate = 50 dollars 36 litres Answer: The price per litre is about $1.389/litre. unit cost rate 1.388 889 dollars/litre example: A world-class sprinter can run 100 m in 9.8 seconds. At what rate can the sprinter run? Answer in m/s and km/h. unit rate in m/s = 100 metres 9.8 seconds unit rate in km/h 10.008 m 1 s 36.73 69 km/h 10.0 08 m/s 1 km 1000 m 60 s 1 min 60 min 1 h Answer: The sprinter's speed is about 10.0 m/s or 36.73 km/h.
Ch. 8: Day 1 notes Rates Page of On a straight-line graph, its slope is a rate of change. When the amount that y changed is compared to the amount that x changed: o rate = rise run = 6 3 o unit rate = slope = The slope is the rate at which the y-coordinate changes as the x-coordinate changes. y 5 0 5 x example: This graph describes a training run. Describe what d distance from home could have happened during the run. distance (km) 3 1 0 10 0 30 0 50 60 t time (min) From 0 to 10 min: speed = slope = 3 km 10 min From 10 to 15 min: speed = slope = 1 km 5 min From 15 to 0 min: speed = slope = 0 km 5 min From 0 to 60 min: speed = slope = km 0 min 60 min 1 h 60 min 1 h 60 min 1 h 60 min 1 h = 18 km/h = 1 km/h = 0 km/h = 6 km/h example: While driving at 70 km/h, a driver looks down to read a text message for 5 seconds. How far has the car travelled in that time? speed = 70 km 1 h 1000 m 1 km 1 h 60 min 1 min 60 s 19. m/s distance = speed time (19. m/s)(5 s) 97. m Answer: The car travelled about 97 m in 5 seconds.
FOUNDATIONS OF MATH 11 Ch. 8 Day : RATE PROBLEMS SOLVING PROBLEMS THAT INVOLVE RATE example: Gas in Surrey is $1.8/L, while gas in Blaine is $.38/gallon. Where is it cheaper to buy 75 litres of gas and what would be the savings? 1 US gallon 3.79 L and use 1 USD = 1.0 CAD o Convert the Blaine price to CAD/L..38 USD 1gal 1.0 CAD 1USD 1gal 3.79L 1.178 786 CAD/L o Calculate the cost savings in CAD. cost in Surrey = 1.8 CAD/L 75 L = 111 CAD cost in Blaine 1.178786 CAD/L 75 L 88.08971 CAD difference in cost = 111 CAD 88.1 CAD =.59 CAD Answer: It is.59 CAD less to buy 75 litres of gas in Blaine. example: Amelia walks briskly for hours and burns 5 calories. Bruce walks at a slower rate, burning 6 calories in 30 minutes. If Amelia walks for 3 hours, how long will Bruce need to walk in order to burn the same amount of calories as Amelia? o Amelia: unit rate = 5 calories = 7 calories/h h number of calories = 7 calories/h 3 h = 681 calories o Bruce: unit rate = number of hours = 6 calories = 1 calories/h 0.5h 681 calories 5.91935 h 1 calories/h Answer: Bruce will need to walk about 5.5 hours to burn the same number of calories as Amelia.
Ch. 8: Day notes Rate Problems Page of example: The simple interest formula is I = Prt, where I is the amount of interest earned, P is the principal (the starting amount), r is the interest rate, and t is the time; time must be in years for an annual interest rate. Mia invested $3000 for months and earned $10 of simple interest. What was the annual interest rate? o Use the simple interest formula. I = Prt months is years 10 = (3000)r() o Solve for r. Answer: The annual interest rate was %. 10 = 6000r r = 10 6000 r = 0.0 example: The low temperature for a certain day was 5.3 C at 3:30 AM. The temperature then rose steadily that day until the high temperature was as 11.8 C at 5:5 PM. A weather forecaster predicted the same temperature increase rate for the next day, from a low of 7 C at 3:00 AM. Estimate the temperature at 7:00 AM the second day. o Calculate the temperature rate of increase. During the first day: temperature increase = 11.8 C ( 5.3 C) = 17.1 C hours after 3:30 AM = 17:5 3:30 = 1:15 = 1.5 hours temperature rate of increase = 17.1 C 1.5 h = 1. C/h o Calculate the temperature at 7:00 AM on the second day. hours after 3:00 AM = 7:00 3:00 = :00 = hours temperature increase = 1. C/h h =.8 C temperature at 7:00 AM = 7 C +.8 C =. C Answer: The temperature is. C below 0 at 7:00 AM on the nd day.
FOUNDATIONS OF MATH 11 Ch. 8 Day 3: SCALE DIAGRAMS SCALE FOR A REDUCTION DIAGRAM A scale diagram is a drawing in which measurements are proportionally reduced or enlarged from actual measurement. Scale is the ratio of a length on a diagram to the corresponding. The scale factor is k = example: The picture below is an aerial photo of NWSS; the photo is a reduction. What is the scale factor of the photo? How long is NWSS? 100 m o The 100 m length on the photo is measured to be.5 cm, so the scale factor, k = o Find the length of NWSS, L =.5 cm 100 m =.5 cm 10 000 cm k = The school image is 9.5 cm long. 0.000 5 = L = 9.5 cm 0.0005 = 38000 cm 1 m 100 cm = 380 m Answer: NWSS is about 380 m long. = 0.000 5 9.5 cm L
Ch. 8: Day 3 notes Scale Diagrams Page of exercise: What is the scale factor of this apartment floor plan? What are the dimensions of the rooms? Dining Room Den Living Room Kitchen Bath Bedroom 5 m [Answer: living room - 3.6 m by.8 m, dining room - 3.9 m by. m, den - 3.6 m by 1. m, kitchen -.7 m by 1.8 m, bath - 1.8 m by 1.8 m, bedroom - 3.0 m by 3.3 m] SCALE FOR AN ENLARGEMENT DIAGRAM example: The diagram of the wasp is an enlargement. What is the scale factor of this picture? How big is the wasp? o The 1 cm length on the diagram is measured to be cm, so the scale factor, k = cm 1 cm = 1 cm o Find the wasp length, L The wasp image is 6 cm long. k = Answer: The wasp is about 1.5 cm long. = 6 cm L L = 6 cm = 1.5 cm What conclusion can you make about the scale factor of a reduction? What conclusion can you make about the scale factor of an enlargement?
FOUNDATIONS OF MATH 11 Ch. 8 Day : SCALE FACTORS & AREAS OF -D SHAPES The scale factor, k = is the linear scale factor and would be applied to linear dimensions such as length, width, height,... SCALE FACTORS FOR AREA The smaller image is an computer chip at its actual size, a 1 cm square. Beside it is a 5 cm enlargement. The linear scale factor, k = 5 cm 1 cm = 5 The area scale factor is diagram area actual area = 5 cm 1 cm = 5 The area scale factor is diagram area actual area which is also equal to k, the square of the linear scale factor. example: A 6 m by 10 m room is drawn as a 3 cm by 5 cm rectangle on a floor plan. Find the scale factor and the area scale factor of the floor plan to determine the area of a kitchen that is 8 cm on the floor plan. o scale factor, k = = 3 cm 6 m = 3 cm 600 cm = 0.005 o area scale factor = k = (0.005) = 0.000 05 o Find the actual kitchen area, A. k = 30 000 cm 0.000 05 = 1 m 100 cm 1 m 100 cm Answer: The kitchen area is 3 m. A = = 3 m diagram area actual area 8 cm A 8 cm = 30000 cm 0.00005
Ch. 8: Day notes Scale Factors & Areas of -D Shapes Page of exercise: The radius of a special giant-size pizza is twice the radius of a small pizza. How much bigger is the giantsize pizza when compared to a small pizza? x x o Define variable x to represent the radius of the smaller pizza. o Calculate the scale factor of the enlargement k = giant-sized radius small radius = x x = o The area scale factor of the enlargement, k = () = area scale factor = giant-sized area small area = π( x) πx = πx πx = OR Answer: The giant-sized pizza is times the size of the small pizza. exercise: The area of a computer display is 1 in. When the display is projected onto a screen, the image is 5ft. What is the scale factor of the projection? o The area scale factor of the projected image is k = image area display area = 5 ft 1 in 1 in 1 ft 1 in 1 ft = 3600 in 1 in = 5 o The scale factor of the projected image is k = 5 = 5 Answer: The scale factor of the projected image is 5.
FOUNDATIONS OF MATH 11 Ch. 8 Day 5: SCALE MODELS & SCALE DIAGRAMS Similar figures are proportionally sized; a two-dimensional objects and all its scale diagrams are similar. These triangles are similar. 3 6 9 6 If the grey triangle is the actual object, then the scale factor for the: the reduction drawing scale factor, k = the enlargement drawing scale factor, k = 8 1 = = 0.5 = 6 = 1.5 Similar figures can also be three-dimensional objects; it and all its scale models are similar. model length The scale factor, k = is the linear scale factor and would be applied to linear dimensions such as length, width, height,... These rectangular prisms are similar because they are proportionally sized. 3 8 6 9 6 1 If the grey prism is the actual object, then the scale factor for the: the reduction model scale factor, k = the enlargement model scale factor, k = model length = = 0.5 model length = 6 = 1.5
Ch. 8: Day 5 notes Scale Models & Scale Diagrams Page of example: A 1:50 die-cast model of the world's largest dump truck is shown. The model is 30.6 cm long, 0.0 cm wide, and 15. cm tall. How long, wide, and tall is the dump truck? o linear scale factor, k = 1 50 = 0.0 k = model length o Find the truck length, L L = 30.6 cm 0.0 = 1530 cm o Find the truck width, W W = 0.0 cm 0.0 = 1000 cm o Find the truck height, H H = 15. cm 0.0 = 770 cm 0.0 = 1 m 100 cm = 15.3 m 0.0 = 1 m 100 cm = 10.0 m 0.0 = 1 m 100 cm = 7.7 m 30.6 cm L 0.0 cm W 15. cm H Answer: The truck is about 15. m long, 10.0 m wide, and 7.7 m tall.
FOUNDATIONS OF MATH 11 Ch. 8 Day 6: SCALE FACTORS & 3-D OBJECTS SCALE FACTORS FOR SURFACE AREA Compare the surface areas of a 1-cm cube and a 5-cm cube. The linear scale factor, k = 5 cm 1 cm = 5 The surface area scale factor is ( ) ( ) = 5 larger surface area smaller surface area = 6 5 cm 6 1 cm The surface area scale factor is model surface area actual surface area which is equal to k, the square of the linear scale factor. SCALE FACTORS FOR VOLUME Compare the volume of a 1-cm cube and a 5-cm cube. The linear scale factor, k = 5 cm 1 cm = 5 The volume scale factor is larger volume smaller volume = 3 3 5 cm 3 1 cm = 53 The volume scale factor is model volume actual volume which is equal to k 3, the cube of the linear scale factor.
Ch. 8: Day 6 notes Scale Factors & 3-D Objects Page of example: A 1:50 die-cast model of a dump truck is shown. The model can carry 100 cm 3 of sand. How much sand will the actual dump truck be able to carry? o linear scale factor, k = 1 50 = 0.0 volume scale factor, k 3 = (0.0) 3 = 0.000 008 o Find the truck load capacity, C C = 3 100 cm 0.000 008 = 150 000 000 cm3 = 150 m 3 k 3 = 0.000 008 = model volume actual volume 3 100 cm C 1 m 100 cm 1 m 100 cm 1 m 100 cm Answer: The dump truck will be able to carry about 150 m 3 of sand. exercise: A cylindrical soup can is 10 cm wide and 10 cm tall. The company wants to make a similar shaped can that will hold twice as much soup. What will the diameter and height of the new can be? o volume scale factor of the can k 3 = o linear scale factor of the can k = 3 1.59 91 o diameter and height of the can, D 1.5991 D 10 cm D 1.59 91 10 cm D 1.599 1 cm Answer: The new can's diameter and height should be about 1.6 cm.