A Rectangular Beam Will Be Cut from a Cylindrical Log $r = 40$

question 17

Multiple Choice

A rectangular beam will be cut from a cylindrical log of radius $r = 40$ inches. Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.

• A) \(\sqrt { 3 }\) in, 80 in
• B) \(\frac { 80 } { \sqrt { 3 } }\) in, \(80 \sqrt { \frac { 2 } { 3 } }\) in
• C) \(\frac { 80 } { \sqrt { 3 } }\) in, \(\frac { 80 } { \sqrt { 3 } }\) in
• D) \(80 \sqrt { \frac { 2 } { 3 } }\) in, \(80 \sqrt { \frac { 2 } { 3 } }\) in
• E) 80 in, \(\sqrt { \frac { 2 } { 3 } }\) in

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Definitions:

Economies Of Scale

The cost advantage achieved by an increase in production, which typically leads to a decrease in the average cost per unit as fixed costs are spread over more units.

Efficiency

The ability to maximize output with minimum input, reflecting optimal allocation and utilization of resources.

Herfindahl Index

A measure of market concentration, calculated by summing the squares of the market shares of all firms in the industry.