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Direct Variation with the Square of a Variable
When a variable varies directly with the square of another variable , the relationship is expressed as:
This extends the basic direct variation formula by replacing the first power of with . The constant is still called the constant of variation and is found the same way: substitute a known pair of values into the equation and solve for . Once is determined, the equation can be used to find for any other value of . Because the variable is squared, the effect of changes in is amplified — for example, doubling will quadruple (since ), rather than merely doubling it as in ordinary direct variation.
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Direct Variation with the Square of a Variable
Maximum Load Supported by a Beam: Direct Variation with the Square
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An operations manager is modeling the relationship between electricity usage and machine run-time. If the usage varies directly with the run-time, what is the effect on the total electricity usage if the machine run-time is exactly tripled?
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Learn After
A safety officer is modeling how the impact force (F) of a falling object varies directly with the square of its impact velocity (v). Which equation correctly represents this relationship using k as the constant of variation?
In a logistics safety manual, the stopping distance of a forklift varies directly with the square of its speed. According to this mathematical relationship, if the speed of the forklift is doubled, the stopping distance will increase by a factor of ____.
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In professional fields such as logistics and safety engineering, researchers often model relationships where one value varies directly with the square of another (such as the relationship between speed and stopping distance). Match each part of this mathematical model with its correct name or functional description.
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In the mathematical model for direct variation with the square of a variable (), the constant of variation () is required to be a non-zero value.
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A technical specification states: 'The pressure load (P) on a structural beam varies directly with the square of the wind velocity (v).' Which mathematical expression correctly represents this relationship using 'k' as the constant of variation?