It is possible to construct a cube of twice the volume of a given cube using only a straightedge and compass.A.TrueB.FalseSUBMITarrow_backPREVIOUS

Break Down the Problem

1. Understanding volume of a cube: The volume $V$ of a cube with side length $s$ is given by $V = s^3$.
2. Finding side length for double volume: To find a new cube with double the volume, we need to set up the equation $2s^3 = S^3$, where $S$ is the side length of the new cube.

Relevant Concepts

1. Calculating new side length: If the original cube has side length $s$, the equation we derived gives: $S^3 = 2s^3$ Therefore, the new side length $S$ can be expressed as: $S = \sqrt[3]{2} \cdot s$

Analysis and Detail

1. Constructing $\sqrt[3]{2}$: To construct a cube with twice the volume, we would need to find a way to construct the side length $S = \sqrt[3]{2} \cdot s$ using a straightedge and compass.
2. Geometric construction: A standard straightedge and compass construction can achieve linear and specific polynomial roots, such as square roots or certain fourth roots, but cannot achieve constructions involving cube roots (like $\sqrt[3]{2}$).

Verify and Summarize

• The construction of the cube root of a number is not possible with just a straightedge and compass due to the limitations of constructible numbers.
• Since constructing $\sqrt[3]{2}$ is not feasible, it follows that constructing a cube of twice the volume is also not possible.

Final Answer

B. False