solid geometry - Volume of a cone, cube, and cylinder

SOURCE:COMPETITION
Number of Problems: 8. : (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 $25\%$ without altering the volume, by what percent must the height be decreased?

$mathrm{(A) } 10 qquad mathrm{(B) } 25 qquad mathrm{(C) } 36 qquad mathrm{(D) } 50 qquad mathrm{(E) } 60$

A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$ ?

$mathrm{(A) } 3qquad mathrm{(B) } 4qquad mathrm{(C) } 5qquad mathrm{(D) } 6qquad mathrm{(E) } 7$

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?

$mathrm{(A) } frac{1}{8}qquadmathrm{(B) } frac{1}{6}qquadmathrm{(C) } frac{1}{4}qquadmathrm{(D) } frac{1}{3...$

A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ or $C$ , as shown. What is the area of quadrilateral $ABCD$ ?

$mathrm{(A)} frac{sqrt{6}}{2}qquadmathrm{(B)} frac{5}{4}qquadmathrm{(C)} sqrt{2}qquadmathrm{(D)} frac{5}{8}qq...$

One dimension of a cube is increased by $1$ , another is decreased by $1$ , and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube?

$extbf{(A)} 8 qquad extbf{(B)} 27 qquad extbf{(C)} 64 qquad extbf{(D)} 125 qquad extbf{(E)} 216$

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?

$extbf{(A)} 7 qquad extbf{(B)} 8 qquad extbf{(C)} 10 qquad extbf{(D)} 12 qquad extbf{(E)} 15$