Unit 14 Review. To be eligible to retake, this packet must be completed in its entirety by the start of class tomorrow!

Transcription

1 Name: Geometry Pd. Unit 14 Review Date: To be eligible to retake, this packet must be completed in its entirety by the start of class tomorrow! Need to break up the figure into triangles Steps: 1. Calculate the apothem! 2. If not already there, draw in the Apothem (mark the right angles) and bisect the central angle to find the vertex angle of the small RIGHT triangle. 3. Bisect the base of the isosceles triangle to find the length of one side of the right triangle. 4. Use SOH CAH TOA to calculate the length of the apothem. 5. Find the area of each triangle 6. Multiply the area by the # of triangles in the regular polygon. Cylinders Formed by translation Base shape: circle (parallel and congruent shapes in the 3D figure) Lateral View: Rectangle (cross section perpendicular to the base) Base View: Circle (cross section parallel to the base) Prisms Formed by translation Polygonal bases Named by their base shapes Lateral View: Rectangle (cross section perpendicular to the base) Base View: same as base shape (cross section parallel to the base) Pyramid Formed by translation and dilation One base that is a polygon Named by base shape Lateral edges congruent *Cylinders can be formed by rotations Cone Formed by rotation of triangle Slant height: height from edge of base to top Base shape: circle (parallel and congruent shapes in the 3D figure) Lateral View: Triangle (cross section perpendicular to the base) Base View: Circle (cross section parallel to the base) Sphere Formed by rotation of circle or semi-cirlce Lateral View: Circle (cross section perpendicular to the base) Base View: Circle (cross section parallel to the base) The great circle: largest circle within a sphere; same diameter as sphere

2 Volume of Prisms and Cylinders: V = Bh (B = area of the base; h is the height/depth of the prism/distance between the bases) Cavalieri s Principle: two of the same solid that have the same base area and the same height, also have the same volume. V (pyramid/cone) = 1 3 Bh V (sphere) = 4 3 πr3 Population Density A ratio of the amount of a population that exists over a given area Population Density = population total area Density of a 3D Solid A ratio that compares an object s weight (mass) to the amount of space (volume) it takes up Mass Density = Volume Find your 3D Figures Cheat Sheet that was given to you!! Station # 1: Properties, Formations, and Cross Sections of Solids 1. Name the solid that has a triangular cross-section parallel to the base and a triangular cross-section perpendicular to the base? 2. Name the solid that is formed by rotating a right triangle around one of the legs? 3. What solid is formed by translating a circle into 3 dimensions? 4. What solid has a cross section parallel to the base that is a trapezoid and a cross section perpendicular to the base that is a rectangle? 5. Describe the cross section of a right circular cone cut through the vertex and perpendicular to the base. 6. Describe the shape of the cross section of a triangular prism cut by a plane parallel to a base. *Note: You will be responsible for any parallel/perpendicular cross section for all of the solids we have studied.

3 Station # 2: Basic Volume 1. Find the volume 2. A regular pyramid is shown at the right. Find the volume of the pyramid to the nearest cubic unit. 3. The radius of a sphere is 6 feet. Find the volume of the sphere in terms of π. Station #3: Equal Volume and Cavalieri s Principle 1. A rectangular prism has an altitude of 15 inches and a base area of 31.2 sq in. A second rectangular prism has a square base, an altitude of 15 in, and the same volume as the first prism. Find the side of the square to the nearest tenth of an inch. 2. A sphere of radius 1 has its volume equal to the volume of a right circular cylinder whose altitude is 3. Find the length of the radius of the cylinder, leave your answer as a fraction.

4 3. Explain why the two stacks of quarters would have the same volume. Station # 4: Further Applications 1. A pile of sand dumped by a hopper is cone shaped. If the radius of the base is 9 feet, and the height is 8 feet, how many cubic feet of sand are in the pile, in terms of π? 2. Soda is sold in aluminum cans that measure 6 inches in height and 8 inches in diameter. How many cubic inches of soda are contained in a full can? (Round answer to the nearest tenth of a cubic inch.) 3. Toughy! Take your time! Station #5 Multiple Step Solids Volume and Density and Surface Area 1. Find the volume of the composite solid. Round to the nearest thousandth.

5 2. 3. Find the volume of the right cylinder. Round your answer to the nearest whole number 4. A 3-inch by 5-inch index card is rotated around a horizontal line and a vertical line to produce two different solids. Which solid has a greater volume? Show your work to defend your answer. 5. The Great Blue Hole is a cylindrical trench located off the coast of Belize. It is approximatgely 1000 feet wide and 400 feet deep. About how many gallons of water does the Great Blue Hole Contain? (1 ft gallons).

6 6. A fish tank in the shape of a rectangular prism has dimensions of 14 inches, 16 inches, and 10 inches. The tank contains 1680 cubic inches of water. What percent of the fish tank is empty? 7. The diameter of a basketball is approximately 9.5 inches and the diameter of a tennis ball is approximately 2.5 inches. The volume of the basketball is about how many times greater than the volume of the tennis ball? 8. Danielle has a full container of ice cream which has a density of 0.1 pounds per cubic inch. The container is a sphere which has a radius of 3 inches. Marisa also has a full container of ice cream, which has a density of 0.01 pounds per cubic inch and is contained in a sphere which has a radius of 3 inches. How many more pounds of ice cream does Danielle have than Marisa? Round any decimals to the nearest hundredth.

7 9. A candle holder is to be built of solid metal. The walls and bottom will be 1ft thick. The outer height will be 20 ft. The inner diameter will be 10 ft. To the nearest cubic foot, how much solid metal will be needed for the job? 10. What is the volume left in the cylinder after the shaded cone region is removed in terms of Pi? 11. A child s toy is fully filled with a heavy liquid in the hemisphere and lighter liquids in the cone and cylinder so that the toy will always right itself (stand up straight) as it is shown in the picture. How much total liquid is contained inside of the toy to the nearest cubic inch?

8 12. Calculate the Surface area and volume of the following solid: 13. a) The surface area of the prism below is 102 cm 2. Find x. b) If 1 can of paint covers 30cm 2, how many cans of paint do you need to buy to cover the prism above? 14. The density of gold is 19.3 grams per cm 3. What is the mass of a gold block that measures 10 cm by 3 cm by 3cm?

9 15. Two containers show below hold candies of the same size. Container A holds 75 candies and container B holds 160 candies. Given the dimensions below, which container has a smaller population density? 16. The diagram below shows a right pentagonal prism. Which statement is always true? a) BC ED b) FG CD c) FJ IH d) GB HC 17. The figure below is a square pyramid. Which of the following is NOT a cross section from the pyramid? 17. Find the area of the regular polygon to the nearest 10 th. Remember to find the apothem first!