Correct option is B
Given:
· A wooden log with a cylindrical shape.
· A beam with a square cross-section is to be cut from the cylindrical log.
· We need to determine the largest fraction of the wood by volume that can be fruitfully utilized as the beam.
Approach:
To maximize the volume of the square cross-section beam that can be cut from the cylindrical log, we need to ensure that the square fits perfectly within the circular cross-section of the cylinder.
Consider the geometry:
The log has a circular cross-section with radius "r."
The beam has a square cross-section with side length "s."
Square inscribed in a circle:
For the largest square that fits inside the circle, the diagonal of the square is equal to the diameter of the circle.
The diagonal of the square is given by: Diagonal =
The diagonal of the square must be equal to the diameter of the circle, which is 2r. Therefore, √2 × s = 2r, and solving for "s" gives: s = r√2.
Volume of the cylindrical log:
The volume of the cylinder is: Volume of cylinder = π × r² × h, where "h" is the height of the cylinder.
Volume of the square beam:
The volume of the square beam is: Volume of beam =
Fraction of the wood utilized:
The fraction of the wood used by the beam is the ratio of the volume of the beam to the volume of the cylinder: Fraction = (Volume of beam) / (Volume of cylinder)
Simplifying this gives: Fraction = 2 / π ≈ 0.6366, which is approximately 64%.
Final Answer:
The largest fraction of the wood by volume that can be fruitfully utilized as the beam is approximately 64%.
Correct Option:(b) 64%.
