Question

question 4 of 10

constructing a cube with double the volume of another cube using only a straightedge and compass was proven possible by advanced algebra.

a. true
b. false

Explanation:

Analyze the construction problem

The statement asks whether constructing a cube with double the volume of another cube using only a straightedge and compass was proven possible by advanced algebra. This is the classic Greek geometry problem known as "doubling the cube" or the Delian problem.

Evaluate the algebraic proof

Using Compass and Straightedge Construction rules, we can only construct lengths that are roots of quadratic equations (degree \(2^k\)). Doubling a cube of unit volume requires constructing a side length of \(\sqrt[3]{2}\). In 1837, Pierre Wantzel used field theory to prove that \(\sqrt[3]{2}\) is not constructible because its minimal polynomial over the rationals, \(x^3 - 2\), has degree 3, which is not a power of 2. Thus, advanced algebra proved this construction is impossible, making the statement false. This is one of the famous Impossible Problems of antiquity.

Answer:

• A. True
• B. False (Correct answer)