# Ergodic Theory - Introductory Lectures by P. Walters

By P. Walters

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Glencoe/McGraw-Hill 40 MathMatters 3 Name _________________________________________________________ EXTRA PRACTICE Date ____________________________ 5-7 VOLUME OF THREE-DIMENSIONAL FIGURES ᮃ EXERCISES Find the volume to the nearest whole number. 14 for ␲. 1. 2. 24 in. 12 in. 5 ft 8 ft 5 ft 3. 4. 5 cm 16 m 14 m 5. How many cubic meters of water can a water tank hold, if the tank is a cylinder 10 m high and 12 m in diameter? 6. 2. What is the height of the prism? 7. A sphere has a radius of 4 m.

X ϩ y Յ 3 yՅ3 xՆ0 yՆ0 maximum P ϭ 3x ϩ y maximum P ϭ 4x Ϫ y Ά 3. x ϩ 2y Ն 8 yՅ1 xՆ0 yՆ0 minimum P ϭ x ϩ 2y Ά Ά 6. x Ϫ y Յ 4 xϩyՅ4 yՆ0 yՆ0 maximum P ϭ 4x ϩ y maximum P ϭ 4x ϩ y Ά 8. Ϫx Ϫ y Ն Ϫ10 yՅ8 xՅ6 xՆ0 yՆ0 Ά 7. x ϩ y Յ 6 yՅ4 xՅ3 xՆ0 yՆ0 minimum P ϭ 2x ϩ 2y maximum P ϭ 6x ϩ y Ά 9. 2x ϩ y Յ 4 xϩyՆ2 yՆ0 10. 4x Ϫ 2y Ն 8 xϪyՅ4 xՅ5 yՆ0 xՆ0 maximum P ϭ 2x Ϫ 2y © Glencoe/McGraw-Hill yՆ0 minimum P ϭ x ϩ 3y 5. 2x ϩ 2y Յ 6 xϩyՆ2 xՆ0 Ά yՆ0 minimum P ϭ 3x ϩ 2y 50 MathMatters 3 Name _________________________________________________________ EXTRA PRACTICE Date ____________________________ 7-1 RATIOS AND PROPORTIONS ᮃ EXERCISES Is each pair of ratios equivalent?

Square pyramid 6. pentagonal prism 7. sphere 8. oblique cylinder 10. oblique rectangular prism 9. hexagonal pyramid © Glencoe/McGraw-Hill 39 MathMatters 3 Name _________________________________________________________ Date ____________________________ 5-6 EXTRA PRACTICE SURFACE AREA OF THREE-DIMENSIONAL FIGURES ᮃ EXERCISES Find the surface area of each figure. Assume that all pyramids are regular pyramids. 14. Round answers to the nearest whole number. 1. 2. 19 m 8 in. 4 in. 6 in. 6m 3. 4. 5 m 18 ft 5.