Lesson 7.3 Understanding the Pythagorean Theorem and Solids
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1 Lesson 7.3 Understanding the Pythagorean Theorem and Solids For this practice, you may use a calculator. Use 3.14 as an approimation for π. Round your answer to the nearest tenth where necessary. For each solid, find the value of the variable. 1.. a in. 4 ft 5 ft 5 in. 1 in. b ft 3. 5 cm 30 cm cm 4. y m 1. m Etra Practice Course 3B 13
2 Solve. Show your work. Round your answer to the nearest tenth. 5. Fred bought a paperweight in the shape of a square pyramid, as shown in the diagram. Find the total surface area of the paperweight. 4 in. 6 in. 6. Helen made a conical lampshade out of a piece of cardboard. She wants to paint the surface of the lampshade. Find the surface area of the lampshade, given the dimensions in the diagram. 50 cm 7. A cone was placed inside a cylindrical container. The cone and the container have the same base, 50 inches, and the same height, 40 inches. Find the slant height of the cone. 0 cm 40 in. 50 in. 14 Chapter 7 Lesson 7.3
3 8. A rectangular bo measures 30 centimeters long, 0 centimeters wide, and 15 centimeters high, as indicated by the diagram. A spider sits at point A while a fly lands at point N. Point N is the midpoint of HG. To catch the fly, the spider crawls from point A to point M, a point directly below the fly, before climbing up to the fly. a) Find the distance from point A to point M. E A H D 30 cm N M F B G 15 cm C 0 cm b) Find the shortest distance between the fly s initial position and the spider s initial position. 9. A composite solid is made up of a cone and a cylinder. The diameter of the cylinder is 3 feet and the slant height of the cone is 3 feet. The cone and the cylinder have the same height. Find the height of the composite solid. 3 ft 3 ft Etra Practice Course 3B 15
4 10. Figure A shows a conical paper cup with a height of 9 inches and a slant height of 10 inches. The cup is cut along a slant edge. Figure B shows the cup spread open on a flat surface. Find the perimeter of Figure B. 10 in. 9 in. O Figure A O Figure B 11. The diagram shows a square pyramid with height 4.5 meters. The length of each slant edge is 5 meters. a) Find the length of a diagonal of the base. 4.5 m 5 m b) Find the area of the base of the pyramid. 16 Chapter 7 Lesson 7.3
5 7. a) AP km PQ km QB 5 4 km Route via P and Q km AR km RB km Route via R km The shorter route is the route that takes him through towns P and Q. b) Difference between the two routes km The difference in the distances of the two routes is 0.7 kilometers. 8. a) Distance from boat to lighthouse RB units Actual distance from boat to lighthouse mi The boat is approimately 4.3 miles from the lighthouse. b) 0 min h Distance 5 Speed Time 4.3 Speed Speed Speed mi/h The speed of the boat is 1.9 miles per hour. Lesson a a a a 169 a b 4 5 b b b 49 b 49 b 7 b 5 7 b Diameter of cylinder cm ,000 1, Radius of sphere m y y y y 0.7 y 0.8 Etra Practice Course 3B 163
6 5. Let the side of the square base be inches ? 5 10? 5 0 Surface area of the pyramid 5 0? 0 1 4? 1? 0? ,360 in The total surface area of the pyramid is 1,360 square inches. 6. Radius of cone 5 1? cm Let the slant height of the cone be centimeters , ,600 5, Curved surface area of cone 5 p? r? 3.14? 10? ,601.1 cm The surface area Helen has to paint is approimately 1,601.1 square centimeters. 7. Radius of cone 5 1? Let the slant height of the cone be s inches. s s ,600 s 5,5 s 5,5 s 47. The slant height of the cone is approimately 47. inches. 8. a) AM 5 AD 1 DM AM AM AM 5 65 AM 5 65 AM 5 5 cm The distance from point A to point M is 5 centimeters. b) The shortest distance between the fly s initial position and the spider s initial position is represented by AN. AN 5 AM 1 MN AN AN AN AN AN 9. cm The shortest distance between the fly s initial position and the spider s initial position is approimately 9. centimeters. 9. Radius of cone 5 1? ft Let the height of the cone be feet Height of composite solid? ft The height of the composite solid is approimately 5. feet. 10. Let the radius of the cone be r inches. r r r r 5 19 r 5 19 r Circumference of base of cone 5? p? r? 3.14? in. Perimeter of Figure B in. The perimeter of Figure B is approimately 47.4 inches. 164 Answers
7 11. a) Let the length of the diagonal of the base be meters ? 5? The length of the diagonal of the base is meters. b) Let the side of the square base be y meters. So, the area of the square is y square meters. y 1 y 5 y y y y 9.5 The area of the base of the pyramid is approimately 9.5 square meters. Lesson a) Let the unknown side of the base of the triangular prism be inches Volume of prism 5 1? 15? 0? in 3 Volume of cylinder 3.14? 3.5? in 3 Volume of composite solid ,138.5 in 3 The volume of the composite solid is approimately 1,138.5 cubic inches. b) Let the height of the cone be feet. Radius of cone 5 1 ft Volume of cylinder 3.14? 1? ft 3 Volume of cone 1 3? 3.14? 1? ft 3 Volume of composite solid ft 3 The volume of the composite solid is approimately 19.8 cubic feet.. Let the radius of the hemisphere be inches Volume of hemisphere 1? 4? 3.14? in 3 Radius of cylinder in. Volume of cylinder 3.14? 10.07? 1 3,80.94 in 3 Volume of solid 5 3, ,081. in 3 The volume of the solid is approimately 3,081. cubic inches. 3. Radius of cone 5 10 cm a) Let the height of the cone be centimeters The height of the cone is 4 centimeters. Etra Practice Course 3B 165
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