When Two Railroad Tracks Merge,the Overlapping Portions of the Tracks $$\frac { x } { 2 } = 2 r \sin ^ { 2 } \frac { \theta } { 2 }$$

question 40

Multiple Choice

When two railroad tracks merge,the overlapping portions of the tracks are in the shapes of circular arcs (see figure) .The radius of each arc r (in feet) and the angle θ are related by​ $$\frac { x } { 2 } = 2 r \sin ^ { 2 } \frac { \theta } { 2 }$$ ​ Write a formula for x in terms of cos θ.​ ​

• A)​ \(x = r \left( 1 - \cos \frac { \theta } { 2 } \right)\)
• B)​ \(x = r ( 1 - \cos \theta )\)
• C) \(x = 2 r ( 1 - \cos \theta )\) ​
• D) \(x = 2 r ( 1 + \cos \theta )\) ​
• E) \(x = 2 r \left( 1 - \cos \frac { \theta } { 2 } \right)\) ​

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Definitions:

Conditional Equation

An equation that is true for certain values of its variables, often used in algebra to describe a relationship subject to specific conditions.

Identity Equation

An equation that is true for all values of the variable, typically used to express fundamental algebraic properties.

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