Imagination is more important than knowledge... Albert Einstein
Guess is more important than calculation --- Knowhowacademy.com
SOURCE:COMPETITION Number of Problems: 8. FOR PRINT ::: (Book)
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?
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 .
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 ?
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,
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
We can break the octahedron into two square pyramids by cutting it along a plane perpendicular to one of its internal diagonals. The cube has edges of length 1 so all edges of the regular octahedron have length . Then the square base of the pyramid has area . We also know that the height of the pyramid is half the height of the cube, so it is . The volume of a pyramid with base area and height is so each of the pyramids has volume . The whole octahedron is twice this volume, so .
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 ?
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 .
One dimension of a cube is increased by , another is decreased by , and the third is left unchanged. The volume of the new rectangular solid is less than that of the cube. What was the volume of the cube?
Let the original cube have edge length . Then its volume is . The new box has dimensions , , and , hence its volume is . The difference between the two volumes is . As we are given that the difference is , we have , and the volume of the original cube was .
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?
Imagine 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 .
We 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 the central cube is included in each of these three cuts. To get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice.
Hence the total volume of the cuts is .
We 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.
.