Solve each of the following problems. The distance that a...

Solve each of the following problems.
The distance that a freely falling body falls varies directly as the square of the time it falls. If a body falls 144 feet in 3 seconds, how far will it fall in 5 seconds?

Key Concepts

Direct Variation

Direct variation is a relationship where one quantity increases or decreases in direct proportion to another. This means that if one quantity is multiplied by a constant, the other quantity is scaled by the same factor, making it possible to describe the relationship with a constant multiplier.

Constant of Proportionality

The constant of proportionality is the fixed factor that links two variables in a directly proportional relationship. It is the multiplier that, when applied to the independent variable, gives the value of the dependent variable, allowing for the prediction of one variable based on the other.

Quadratic Relationships

A quadratic relationship occurs when one quantity varies as the square of another. This means that if the independent variable is scaled by a factor, the dependent variable is scaled by the square of that factor. This type of relationship is common in many physics problems and mathematical models, particularly where geometric growth is involved.

Proportional Reasoning

Proportional reasoning involves setting up and solving ratios to determine unknown values by understanding the multiplicative relationships between quantities. This technique is fundamental in solving problems where one quantity is directly linked to another, whether linearly or through more complex relationships such as squares or higher powers.

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