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A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by without altering the volume, by what percent must the height be decreased? ' |
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A wooden cube units on a side is painted red on all six faces and then cut into unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is ? ' |
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Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron? ' |
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A cube with side length is sliced by a plane that passes through two diagonally opposite vertices and and the midpoints and of two opposite edges not containing or , as shown. What is the area of quadrilateral ? ' |
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A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? ' |
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When the diameter is increased by , it is increased by , so the area of the base is increased by . To keep the volume the same, the height must be of the original height, which is a reduction . |
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Since there are little faces on each face of the big wooden cube, there are little faces painted red. Since each unit cube has faces, there are little faces total. Since one-fourth of the little faces are painted red, |
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We can break the octahedron into two square pyramids by cutting it along a plane perpendicular to one of its internal diagonals. |
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Since , it follows that is a rhombus. The area of the rhombus can be computed by the formula , where are the diagonals of the rhombus (or of a kite in general). has the same length as a face diagonal, or . is a space diagonal, with length . Thus . |
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Let the original cube have edge length . Then its volume is . |
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Solution 1Imagine making the cuts one at a time. The first cut removes a box . The second cut removes two boxes, each of dimensions , and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is . Therefore the volume of the rest of the cube is . Solution 2We can use Principle of Inclusion-Exclusion to find the final volume of the cube. There are 3 "cuts" through the cube that go from one end to the other. Each of these "cuts" has cubic inches. However, we can not just sum their volumes, as Hence the total volume of the cuts is . Therefore the volume of the rest of the cube is . Solution 3We can visualize the final figure and see a cubic frame. We can find the volume of the figure by adding up the volumes of the edges and corners. Each edge can be seen as a box, and each corner can be seen as a box. |