STATE MU ALPHA THETA 2008 GEOMETRY 3-D

Transcription

1 STATE MU ALPHA THETA 2008 GEOMETRY 3-D 1) In terms of π, what is the volume of a sphere with a radius of 6? a) 36π b) 48π c) 288π d) 324π 2) In terms of π, what is the total surface area of a circular cylinder with a diameter of 8 and a height of 4? a) 72π b) 64π c) 108π d) 36π 3) What is the volume of a right rectangular prism with a length of 4, a width of 6, and a height of 10? a) 108 b) 144 c) 208 d) 240 4) In terms of π, what is the total surface area of a sphere with a radius of 12? a) 648π b) 472π c) 76π d) 48π ) What is the volume of a cube with an edge of 8? a) 488 b) 16 c) 496 d) 08 6) A x8 rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction. a) 2 b) 2 40 c) 13 d) 8 7) Eighty-five unit cubes are placed in a line such that they are joined face to face. Find the number of square units the surface area of the resulting solid. a) 342 b) 36 c) 346 d) 38

2 8) A right circular cone with altitude 8 and slant height 10 is attached to one end of a cylinder with height 10. A hemisphere with the same circumference as the cylinder is attached to the other end of the cylinder. If the cylinder and the cone have the same radius, find the number of cubic units in the total volume of the solid. a) 288π b) 600π c) 300π d) 120π 9) A= the number of faces a cube has. B= the number of vertices a tetrahedron has. C= the total surface area of a cube with an edge length of 2. Find the value of (A B) + C. a) 36 b) 48 c) 24 d) 16 10) A cubical box with 10 inch-long sides contains a sphere with diameter of ten inches. What is the maximum number of inches that any point on the sphere can be from any point on the surface of the cube? Express your answer as a decimal to the nearest hundredth. a) b) c) d) ) A dodecahedron has twelve pentagonal faces. A face diagonal is a line segment that connects two nonadjacent vertices and lies on a face. How many faces diagonals does a regular dodecahedron have? a) 64 b) 6 c) 60 d) 48 12) United Parcel Service (UPS) requires that the sum of a package s length, x, and girth, 2y + 2z, where y is the width and z is the height, the girth not to exceed 130 inches. If a package has a width of 20 inches and a length of 36 inches, what is the maximum number of inches in the height of the package if UPS is to accept it? a) 24 b) 28 c) 26 d) 27

3 13) A regular icosahedron has 20 triangular faces. A polygon is inscribed inside it such that each vertex of the polygon is the centroid of a face of the icosahedron. What is the name of the inscribed polygon? a) Cube b) Octahedron c) Dodecahedron d) Tetrahedron 14) A spherical scoop of vanilla ice cream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the ice cream spreads out uniformly forming a cylindrical region 8 inches in radius. Assuming the density of the ice cream remains constant, how many inches deep is the melted ice cream? Express your answer as a common fraction. a) 3 1 b) 8 c) 7 4 d) 4 3 1) Three faces of a cube are randomly selected. What is the probability that they share a common vertex? a) 9 b) 7 4 c) 2 1 d) 6 16) The area of one face of a cube is 9 square centimeters. If the length of each edge is increased by 1 cm, what is the number of cubic centimeters in the volume of the resulting cube? a) 36 b) 24 c) 72 d) 64 17) Cone A has a radius of 14.8 and a height of 28.3 and cone B has a radius of 28.3 and a height of What is the ratio of the volume of cone A to the volume of cone B? Express your answer as a common fraction. 129 a) b) c) d) 283

4 18) A pipe with inside diameter 10 inches is to carry water from a reservoir to a small town in an arid land. Neglecting the friction and turbulence of the water against the inside of the pipes, what is the minimum number of 2 inchinside-diameter pipes of the same length needed to carry the same volume of water to the arid town? a) 30 b) 2 c) 20 d) 3 19) A Convex 42-sided polygon has 24 vertices. How many edges does it have? a) 18 b) 20 c) 24 d) 28 20) How many square centimeters are in the surface area of the rectangular solid with a length of 2 cm, a width of 3 cm, and a height of 4 cm? a) 2 b) 4 c) 48 d) 24 21) The outside faces of a large cube are painted. The large cube is then divided into unit cubes. There are exactly 64 unit cubes that have no paint. How many unit cubes were used to create the large cube? a) 208 b) 212 c) 216 d) ) How many square centimeters are in the total surface area of a cube whose edge length is 9.3 cm? Express your answer as a decimal to the nearest hundredth. a) b) c) d) ) The surface of a cube is painted red and then cut into unit cubes. How many of the unit cubes have exactly one red face? a) 18 b) 24 c) 16 d) 28 24) The basement of a new home will be 30 feet long, 24 feet wide and 9 feet deep. If the earth is hauled away in loads of 3 cubic yards each, how many loads must be hauled away? a) 72 b) 84 c) 76 d) 80

5 2) When a rock is submerged in water in a cylindrical container with a radius of 3 inches, the water level rises 1. inches. What is the number of cubic inches in the volume of the rock? Express as a common fraction in terms of π. 29π a) 29π b) 2 c) 27π d) 24π 26) An ice cream cone with a diameter of 3 inches and a height of 6 inches is filled to a depth of 3 inches with ice cream, resulting in a cone with a diameter of 1. inches. What is the number of cubic inches of ice cream that would be needed to fill the remainder of the cone to the top? Express your answer as a decimal number to the ten-thousandths times π. a) 3.948π b) 3.864π c) π d) 3.937π 27) A solid cube measures 21 cm on edge. Nine cubes of edge 3 cm are removed from the center of each face of the original cube. What is the number of square centimeters in the surface area of the new object? a) 3,294 b) 3,249 c) 3,394 d) 3,349 28) A 3 inch by 4 inch rectangle is rotated about a corner. What is the maximum number of square inches in the area of the region touched by some point of the rectangle as it makes a full rotation? Express your answer in terms of π. a) 20π b) 18π c) 16π d) 24π 29) Stephanie, Julian, Patrick and Alexia are participating in a cake contest. Julian s cake is in the shape of a rectangular prism with a length of 7, a width of 4, and a height of 6. Stephanie s cake is in the shape of cube with an edge length of 6. Patrick s cake is in the shape of a circular cylinder radius of 2 and a height of 3. Alexia s cake is in the shape of a sphere with a radius of 3. Whose cake has the largest volume? a) Julian b) Stephanie c) Alexia d) Patrick

6 30) A cube has a volume of 64 cubic units. How many units are in the sum of the lengths of all its edges? a) 48 b) 36 c) 6 d) 64 31) An edge of cube C is 4 centimeters in length. The number of cubic centimeters in the volume of C equals the number of square centimeters in the area of square S. How many centimeters are in the perimeter of S? a) 32 b) 36 c) 42 d) 46 32) The Tastee Treats Ice Cream Shoppe presently offers the ice cream in cuboid containers with dimensions of 18 cm x 12 cm x 10 cm. The packaging firm with which they work indicates to them that a cylinder with a radius of 7. cm and a height of 12.2 cm would decrease the surface area of the container and would be cheaper. To the nearest square centimeter, what is the positive difference between the numbers of square centimeters in the surface areas of the two containers? Use 3.14 as an approximation of π. a) 104 b) 102 c) 112 d) ) A cylindrical quarter has a inch diameter and a inch height. What would be the number of inches in the height of a coin whose volume is 1 exactly four times that of the given quarter and whose diameter 1 inches? 8 Express your answer as a common fraction. 16 a) b) c) d) ) Blocks of modeling clay are right rectangular prisms six inches by two inches by one inch. How many whole blocks are needed to mold a cylindrical sculpture seven inches high and four inches in diameter? a) 4 b) c) 6 d) 7

7 3) Two ants are crawling on a 4-inch-by-6-inch-by-8-inch brick. What is the farthest, inches, the two ants could get from one another? Express your answer in simplified radical form. a) 4 29 b) 29 c) 3 29 d) ) A cube is increased to form a new cube so that the surface area of the new cube is double that of the original cube. By what factor is the volume of the cube increased? Express your answer is simplified radical form. a) 2 2 b) 3 c) 2 d) ) The volume of a given sphere is 36π cubic inches. How many square inches are in its surface area? Express your answer in terms of π. a) 16π b) 36π c) 24π d) 48π 38) A water tower is in a small South Dakota town and shaped like a cylinder with a hemisphere attached to the bottom face and a cone attached to the top face. The cylinder, hemisphere and cone have a radius of 6 ft and the cone has a height of 8 ft. The total height from the top of the cone to the bottom of the hemisphere is 26 ft. Find the number of cubic feet in the volume of the tank. Express your answer in terms of π. a) 64π b) 648π c) 98π d) 672π 39) Two fuel tanks, one a cylinder and one a cone, are full of jet fuel. The cylindrical tank holds 00 gallons more than the conical one. If 200 gallons are pumped from each tank the cylindrical one will contain twice as much fuel as the conical one. How many gallons of jet fuel does the cylindrical tank contain when it is full? a) 1,200 b) 1,396 c) 1,424 d) 1,080 40) A solid 3 x 3 x 3 cube weighs pounds and it worth $216. How many dollars is a solid 4 x 4 x 4 cube made of the same material worth? a) 412 b) 496 c) 08 d) 36