Chapter 1 Measurement
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1 Chapter 1 Measurement Math Chapter 1 Measurement Sections : Goals: Converting between imperial units by unit analysis Converting between SI units Converting between SI and imperial units Imperial units inch, foot, yard, mile SI units metric system (I) Converting between imperial units by unit analysis How do we set a conversion up to be solved by unit analysis? Example: Using the scales in the table below, use the units only to create a single line where each stated unit is converted to the indicated unit. (a) yards to feet = yds x 1 ft. = 12 in. 1 yd. = 3 ft. 1 yd. = 36 in. 1 mi. = 1760 yd. 1 mi. = 5280 ft. From the scale above, what units must be placed in the numerator (top) and denominator (bottom) so that yards cancel and feet remain?
2 Chapter 1 Measurement Math Sometimes conversions may involve more than one scale 1 ft. = 12 in. 1 yd. = 3 ft. 1 yd. = 36 in. 1 mi. = 1760 yd. 1 mi. = 5280 ft. (b) = mi x x miles to inches From the scale above, what units must be placed in the numerators (top) and denominators (bottom) so that miles cancel and inches remain? Example: Estimate the amount of wire that feeds through the floor joists. (a) length in inches (b) convert to feet Converting by Unit Analysis 1 ft. = 1 =
3 Chapter 1 Measurement Math Relationships between units: Example 1: Convert 100 yd to: (a) ft. (b) inches 1 ft. = 12 in. 1 yd. = 3 ft. 1 yd. = 36 in. 1 mi. = 1760 yd. 1 mi. = 5280 ft. Example 2: A sunken ship is discovered by sonar to be in ft of water. Convert this depth to miles. Example 3: Convert yds to yards, feet and inches. Example 4: The perimeter of a ceiling in a room is measured for the purpose of installing crown molding. The perimeter of the ceiling is 272 in. (a) What is the perimeter of the ceiling in feet? (b) The cost of crown molding is $3.75/ft.. Determine the cost of ordering crown molding to install in the room.
4 Chapter 1 Measurement Math (II) Converting between SI units The smallest metric measurement on the ruler below is the. How many divisions make up 1 cm? Answer: When we use metric measurements for determining length it is based on increments of 10. SI units Abbreviation Relationship between units millimeter mm centimetre cm 1 cm = 10 mm metre m 1 m = 100 cm kilometre km 1 km = 1000 m Example 5: Convert each length to the indicated measurement cm = mm m = km mm = cm m = cm 5. 6 km = m cm = m
5 Chapter 1 Measurement Math (III) Converting Imperial Units to SI units 1 foot = 12 inches 1 yard = 3 feet 1 mile = 1760 yards 1 inch = 2.54 centimetres 2.5 centimetres 1 mile 1.6 kilometres Example 6: Convert each measurement to the nearest tenth. (a) 8 in = cm (b) 264 mi = km (c) 7 ft = cm (d) 1 yd = mm (e) 6 ft. 7 in. = cm (f) 4 yd. 2 ft. 2 in = m Questions: Page #7,8ab,10b,11a,15,17,18 Page # 4,5,6,7a (in part (ii) delete ft and in), 10,12,14,15
6 Chapter 1 Measurement Math REVIEW OF SURFACE AREA Area formulas: (units) 2 Rectangle: A = l w Square: A = l l or A = l 2 Circle: A = π r 2 Triangle: A = bh 2 or A = 1 2 bh Pythagorean Theorem: a2 + b 2 = c 2 Net Diagram the diagram formed if all sides were unfolded Find the surface area of the following: 1. Square/Rectangular Prism (A) 10cm 10cm 10cm (B) 8cm 10cm 6cm
7 Chapter 1 Measurement Math Triangular Prism 4.0cm 6.0cm 10.0cm NOTE: the top/bottom are circles and the curved surface is a rectangle whose length is the circumference of the circle 3. 8 cm 12 cm
8 Chapter 1 Measurement Math Section 1.4: Surface Area of Right Pyramids and Right Cones Pyramids: are named or described by the shape of its base. SA of pyramids = SA of base + SA of lateral sides (triangles) NOTE if the base polygon of a pyramid is regular (all sides =) then all lateral triangles are congruent Example 1: Determine the surface area of the regular triangular pyramid (regular tetrahedron) Sketch the tetrahedron net diagram 13 yd
9 Chapter 1 Measurement Math The Right Pyramid A three dimensional object that has Triangular faces (lateral faces) Base is a polygon (many sides) Point on top is called the apex S represents slant height (the height Of the triangular face) h is the height of the pyramid (the Perpendicular distance from the Apex to the center of the base) The Lateral Area, AL, is the area Of the exposed triangles (surface area without the base) Example 2: Determine the lateral area and the surface area of the square pyramid Sketch the net diagram
10 Chapter 1 Measurement Math Example 3: Determine the slant height of a square pyramid given the surface area. Example 4: Determine the surface area of a square pyramid given the height of the pyramid. Height is 8m and base is 12m
11 Chapter 1 Measurement Math Example 5: Determine the surface area of a rectangular pyramid. Net Diagram: Questions: Page #4,5,8a,10,16b,13b,18
12 Chapter 1 Measurement Math Right Cones: SA = lateral area + base SA = Example 6: Determine the surface area of the right cone. 18cm 8 cm
13 Chapter 1 Measurement Math Example 7: Determining an unknown measure of a cone SA = 2325 cm 2 and radius r = 10 cm. Determine the height h of the cone h 10cm Questions: Page #6a7b,8b,11,15,16a
14 Chapter 1 Measurement Math Section 1.5: Volume of Right Pyramids and Right Cones Volume the amount of space an object occupies Capacity the amount of material a container holds --volume and capacity are measured in cubic units. (units) 3 (A) Right Square Prisms, Right Rectangular Prisms V = A B H 5 cm 3 cm 2 cm The Volume of a Rectangular Pyramid is one-third the volume of the Right Rectangular Prism with same dimensions 5 cm 3 cm 2 cm Example 1: A right square pyramid has a base side length of 6.4 cm. Each triangular face has two equal sides of length 8 cm. Determine the height and volume of the pyramid. 8cm 8cm 6.4 cm 6.4 cm
15 Chapter 1 Measurement Math (B) Right Cylinder 5 cm 16cm Similarly, the volume of a right cone is one third the volume of a right cylinder with the same base and height. 16cm 5 cm Example 2: Determine the volume for: 5 cm 4 cm
16 Chapter 1 Measurement Math Example 3: The height of the right cone below is 10 cm and its volume is 377 cm 3 Determine the diameter. 10 cm Questions: Page #4a,5b,6a,7b,8,9,11,15,18bd Section 1.6: Surface Area and Volume of a Sphere: A sphere is the set of points in space that are the same fixed distance from a fixed point, which is the center. Sphere: Hemisphere: Surface Area: SA = 4πr 2 Surface Area: SA = 2πr 2 + πr 2 = 3πr 2 Volume: v = 4 3 πr3 or v = 4πr3 3 Volume: v = 2 3 πr3 or v = 2πr3 3
17 Chapter 1 Measurement Math Example 1: An official basketball has a radius of 12.5 cm and usually has a leather covering. Approximately how much leather, in cm 2, is required to cover 12 official basketballs? Example 2: The surface area of a softball is approximately in 2. Determine the diameter of the softball. Example 3: Find the surface area of the hemisphere that has a radius of 8.0cm Example 4: A fitness ball when inflated with air has a circumference of 198 cm. Determine the volume of the fitness ball to the nearest tenth of a cm 3. Questions: Page 51 #3d,4a,5,8,11,18,19,20
18 Chapter 1 Measurement Math Section 1.7 Solving Problems Involving Objects Example 1: 2 ft 4 ft (A) Find Surface Area (B) Find Volume
19 Chapter 1 Measurement Math Example 2: Below is a sketch of a grain bin. If a farmer's grain truck can hold 560 cubic feet of barley, how many truckloads of barley are required to fill the bin? Example 3: Determine the surface area and volume of a right cylinder with a right cone of the same height removed. Questions: Page #3-11 (omit #4)
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