The surface area of a cube is 6 times the area of each square face of the cube (SA = 6s²), or 54 = 6s². So each side is 3.The diagonal of the square forms the hypotenuse of a right triangle.Remember that the hypotenuse of a right triangle is always longer than either leg.Therefore, the diagonal is larger than 3.

First, write out your formulas and draw a figure.The volume of a cube is V = s³ = 512, giving you s = 8.The surface area of a cube is 6 times the area of each square face of the cube (SA = 6s²), therefore, 6 × 8² = 384.

You will be subtracting the volume of the cylinder from that of the cube, so the answer will contain π; eliminate choices (A) and (E). To find the volume of the figure, start with the volume of the cube: V= s³ = 9³ = 729.The formula for volume of a cylinder is V = πr²h.The cylinder runs the length of the cube, so its height is the same as the length of the cube’s edge, 9. Next, find the radius.The length of diagonal AB is 9√2(remember your special triangles—this is a 45-45-90 triangle!).The points between A and B are equally spaced, so the length of CD, the circle’s diameter, is ¹⁄₃ the length of AB,3√2.The radius is 1/2 the diameter, or \(\frac{3\sqrt{2}}{2}\). Plug the radius and the height into the formula:\(V=\pi r^{2}b=\pi \left ( \frac{3\sqrt{2}}{2} \right )^{2}(9)=\frac{81\pi }{2}\). Subtract this from the cube’s volume for a final answer of 729 \(729-729-\frac{81\pi }{2}\).

Draw it! You should end up with two squares oriented the same way touching at C.The squares have the same area, so their sides must be the same length. Plug in a side length for the squares to simplify the comparison—try 2. A square cut in half along its diagonal yields a pair of 45-45-90 triangles, so these two squares with sides of 2 each have diagonals of 2√2. Diagonals AC and CE connect to form segment AE. Quantity A is 3 × 2 = 6, and Quantity B is 2√2 + 2√2 = 4√2 Quantity A is greater.

Plug in a value for n:Try n = 3. If each edge of the cube is 3, AB = DE = 3, and because the diagonal of a square forms two 45-45-90 triangles,BD = AE = 3√2.The total perimeter is 3 + 3 + 3√2 + 3√2 = 6 + 6√2. Now plug in 3 for n in the answer choices; only choice (A) hits your target.

The diagonal of a square is always longer than its side, so half a diagonal—segment AE—must be longer than half a side. Half the length of a side of this square is 4.Therefore, AE is larger than 4.

Three times the surface area of a cube with edge length of 1 cm is three times the area of each square face times the number of faces: 3 × (1 cm × 1 cm) × 6 faces = 18 cm².The surface area of a cube with edge length of 3 cm is (3 cm × 3 cm) × 6 faces = 54 cm². Quantity B is greater.

The answer asks for ¹⁄₃ of the whole volume, so begin by dividing the height of the trunk by 3 to find the volume of one of the sections of the trunk:\(\frac{6\, ft}{3}\) = 2ft.The volume formula for a cylinder is: V= πr²h = π × 2² × 2= 8π.

The original slice is cut from a pizza with a diameter of 10, and therefore a circumference of 10π.This slice represents \(\frac{1.25\pi }{10\pi }=\frac{1}{8}\) of the circumference and therefore ¹⁄₈ of the area,\(\frac{25\pi }{8}\) which weighs 4 ounces. A serving weighs 8 ounces, which covers double the area,\(\frac{25\pi }{8}\).The area of the six pizzas is (6)π3² = 54π. Dividing this by the area of one serving gives you the total number of servings that the six pizzas represent:\(\frac{54\pi }{\left ( \frac{25\pi }{4} \right )}=8\frac{16}{25}=8.64\).The six pizzas yield 8 servings

To solve this one, Plug In for r and m:Try r = 2 and m = 4. If ¹⁄₂ of the pizza has been eaten, and the remaining is divided into 4 equal slices, then each of those remaining ¹⁄₂ pieces ¹⁄₈ is of the whole pizza. Now plug in 2 for r and 4 for m in the answer choices; only choice (B) hits your target answer of ¹⁄₈.