# basic math surface area and volume and surface area formulas math for water technology mth 082...

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• Slide 1
• Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10
• Slide 2
• Objectives Become proficient with the concept of volume as it pertains to common geometric shapes. Solve waterworks math problems equivalent to those on State of Oregon Level I and Washington OIT Certification Exams Become proficient with the concept of volume as it pertains to common geometric shapes. Solve waterworks math problems equivalent to those on State of Oregon Level I and Washington OIT Certification Exams
• Slide 3
• RULES FOR AREA PROBLEMS 1.IDENTIFY THE OBJECT 2.LABEL THE OBJECT 3.LOCATE THE FORMULA 4.ISOLATE THE PARAMETERS NECESSARY 5.CARRY OUT CONVERSIONS 6.USE YOUR UNITS TO GUIDE YOU 7.SOLVE THE PROBLEM 1.IDENTIFY THE OBJECT 2.LABEL THE OBJECT 3.LOCATE THE FORMULA 4.ISOLATE THE PARAMETERS NECESSARY 5.CARRY OUT CONVERSIONS 6.USE YOUR UNITS TO GUIDE YOU 7.SOLVE THE PROBLEM
• Slide 4
• What is surface area? Solid- A 3-D figure (combo of prism, clyinder, cones, spheres, etc.) Total surface area- the sum of the areas of each face of the 3-D solid Lateral surface area- The lateral area is the surface area of a 3D figure, but excluding the area of any bases (SIDES ONLY). It is always answered in square units 2 For example - to find the surface area of a cube with sides of 5 inches, the equation is: Surface Area = 6*(5 inches) 2 = 6*(25 square inches) = 150 sq. inches Solid- A 3-D figure (combo of prism, clyinder, cones, spheres, etc.) Total surface area- the sum of the areas of each face of the 3-D solid Lateral surface area- The lateral area is the surface area of a 3D figure, but excluding the area of any bases (SIDES ONLY). It is always answered in square units 2 For example - to find the surface area of a cube with sides of 5 inches, the equation is: Surface Area = 6*(5 inches) 2 = 6*(25 square inches) = 150 sq. inches
• Slide 5
• What is volume? The amount of space that a figure encloses It is three-dimensional It is always answered in cubed units 3 The amount of space that a figure encloses It is three-dimensional It is always answered in cubed units 3
• Slide 6
• Surface Area of a Sphere A sphere is a perfectly symmetrical, three-dimensional geometrical object; all points of which are equidistant from a fixed point. Sphere Surface Area= 4 r = d A sphere is a perfectly symmetrical, three-dimensional geometrical object; all points of which are equidistant from a fixed point. Sphere Surface Area= 4 r = d m m D=diameter r=radius c c i i r r c c u u f f r r e e n n c c e e r d
• Slide 7
• DRAW: Given: Formula: Solve: DRAW: Given: Formula: Solve: The diameter of a sphere is 8 ft. What is the ft 2 surface area of the sphere? D=8 ft A= d A= 3.14 (8 ft) 2 A= 3.14 (64 ft 2 ) A= 201 ft 2 D=8 ft A= d A= 3.14 (8 ft) 2 A= 3.14 (64 ft 2 ) A= 201 ft 2 1.31.4 ft 2 2.25.1 ft 2 3.201 ft 2 4.628 ft 2 1.31.4 ft 2 2.25.1 ft 2 3.201 ft 2 4.628 ft 2 8 ft
• Slide 8
• Surface Area of a Hemisphere A hemisphere is a sphere this is divided into two equal hemispheres by any plane that passes through its center; A half of a sphere bounded by a great circle. In waterworks its a vat. Hemisphere or Vat Surface Area= 2 r A hemisphere is a sphere this is divided into two equal hemispheres by any plane that passes through its center; A half of a sphere bounded by a great circle. In waterworks its a vat. Hemisphere or Vat Surface Area= 2 r r d r d Vat
• Slide 9
• Volume of a Sphere and Hemisphere Sphere Volume = 4 r = ( d) 3 6 Hemisphere or VAT Volume = (2 ) r 3 3 Sphere Volume = 4 r = ( d) 3 6 Hemisphere or VAT Volume = (2 ) r 3 3 Vat d d hemisphere
• Slide 10
• The diameter of a sphere is 20 ft. What is the ft 3 volume of the sphere? D=20 ft V= 2 (0.785) (D 2 )(D) 3 V= 2 * 0.785*(20 ft) 2 (20 ft) 3 V= (12560 ft 3 ) 3 V= 4187 ft 3 D=20 ft V= 2 (0.785) (D 2 )(D) 3 V= 2 * 0.785*(20 ft) 2 (20 ft) 3 V= (12560 ft 3 ) 3 V= 4187 ft 3 1.526 ft 3 2.6583 ft 3 3.4187 ft 3 4.6280 ft 3 1.526 ft 3 2.6583 ft 3 3.4187 ft 3 4.6280 ft 3 20 ft DRAW: Given: Formula: Solve: DRAW: Given: Formula: Solve:
• Slide 11
• Volume of a cone Volume of cone = 1/3 ( r height) = 1/3 ( d height) or (0.785) (D) (height) 3 Volume of cone = 1/3 ( r height) = 1/3 ( d height) or (0.785) (D) (height) 3 A cone is a solid with a circular base. It has a curved surface which tapers (i.e. decreases in size) to a vertex at the top. Cone height is the perpendicular distance from the base to the vertex. http://www.onlinemathlearning.com/volume-formula.html
• Slide 12
• Volume of a cone Calculate the volume of a cone that is 3 m tall and has a base diameter of 2m 2 m 3 m V= 1/3 ( r height) V= 1/3 ( 1m 3m) V= 1/3 ( 3 m 3 ) V=0.33(9.42m 3 ) V=3.14m 3 V= 1/3 ( r height) V= 1/3 ( 1m 3m) V= 1/3 ( 3 m 3 ) V=0.33(9.42m 3 ) V=3.14m 3
• Slide 13
• DRAW: Given: Formula: Solve: DRAW: Given: Formula: Solve: The bottom portion of a tank is a cone. If the diameter of the cone is 50 ft and the height is 3 ft, how many ft 3 of water are needed to fill this portion of the tank? D= 50 ft, h= 3 ft, I know r= 25 ft! V= 1 (0.785)(D 2 )h 3 V= 1(0.785)(50 ft) 2 (3ft) 3 V= (0.785)(2500ft 2 )(3ft) 3 V=(5888ft 3 ) 3 V= 1962ft 3 D= 50 ft, h= 3 ft, I know r= 25 ft! V= 1 (0.785)(D 2 )h 3 V= 1(0.785)(50 ft) 2 (3ft) 3 V= (0.785)(2500ft 2 )(3ft) 3 V=(5888ft 3 ) 3 V= 1962ft 3 D= 50 ft r= 25 ft h=3 ft 1.51032 ft 3 2.3533 ft 3 3.1649 ft 3 4.1962 ft 3 1.51032 ft 3 2.3533 ft 3 3.1649 ft 3 4.1962 ft 3
• Slide 14
• Lateral Surface Area of a Cone Area of cone = 1/2 ( d slant height) = slant height d= diameter
• Slide 15
• Cylinder (TANK OR PIPE!!!) H=height r=radius d=diameter A cylinder is a solid containing two parallel congruent circles. The cylinder has one curved surface. The height of the cylinder is the perpendicular distance between the two bases.
• Slide 16
• Volume of a Cylinder (TANK OR PIPE!!!) Volume = r height = d height Volume= 0.785(diameter 2 )(depth) Volume = r height = d height Volume= 0.785(diameter 2 )(depth) H=height r=radius d=diameter
• Slide 17
• DRAW: Given: Formula: Solve: DRAW: Given: Formula: Solve: What is the capacity of a cylindrical tank in cubic feet if it has a diameter of 75.2 ft and the height is 42.3 ft from the base? D= 75.2 ft, h= 42.3 ft V= 0.785(diameter 2 )(depth) V=(0.785)(75.2 ft) 2 (42.3ft) V= (0.785)(5655ft 2 )(42.3ft) V=(187,778ft 3 ) D= 75.2 ft, h= 42.3 ft V= 0.785(diameter 2 )(depth) V=(0.785)(75.2 ft) 2 (42.3ft) V= (0.785)(5655ft 2 )(42.3ft) V=(187,778ft 3 ) D=75.2 ft H=42.3 ft 1.2500 ft 3 2.188,000 ft 3 3.105,625 ft 3 1.2500 ft 3 2.188,000 ft 3 3.105,625 ft 3
• Slide 18
• DRAW: Given: Formula: Solve: DRAW: Given: Formula: Solve: A pipe is 16 inch in diameter and 550 ft long. How many gallons does the pipe contain? D= 16 in or 1.33 ft, L= 550 ft V= 0.785(diameter 2 )(length) V=(0.785)(1.33 ft) 2 (550 ft) V= (0.785)(1.77ft 2 )(550 ft) V= 764 ft 3 V=(764ft 3 ) (7.48 gal/1ft 3 ) V= 5716 gallons D= 16 in or 1.33 ft, L= 550 ft V= 0.785(diameter 2 )(length) V=(0.785)(1.33 ft) 2 (550 ft) V= (0.785)(1.77ft 2 )(550 ft) V= 764 ft 3 V=(764ft 3 ) (7.48 gal/1ft 3 ) V= 5716 gallons D=16 in l=550 ft 1.4,295 gallons 2.5,716 gallons 3.51,670 gallons 4.7,282 gallons 1.4,295 gallons 2.5,716 gallons 3.51,670 gallons 4.7,282 gallons
• Slide 19
• Surface Area of a Solid Cylinder In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle. You can find the area of the top (or the bottom). That's the formula for area of a circle ( r 2 ). Since there is both a top and a bottom, that gets multiplied by two. The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 r)* h. Add those two parts together and you have the formula for the surface area of a cylinder (www.webmath.com). In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle. You can find the area of the top (or the bottom). That's the formula for area of a circle ( r 2 ). Since there is both a top and a bottom, that gets multiplied by two. The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 r)* h. Add those two parts together and you have the formula for the surface area of a cylinder (www.webmath.com).
• Slide 20
• Surface Area of a Solid Cylinder Surface Area = Areas of top and bottom +Area of the side Surface Area = 2(Area of top) + (perimeter of top)* height Surface Area = 2 r 2 + 2 rh Surface Area = Areas of top and bottom +Area of the side Surface Area = 2(Area of top) + (perimeter of top)* height Surface Area = 2 r 2 + 2 rh H=height r=radius d=diameter
• Slide 21
• Volume of wa

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