Solve the variation problems in Exercises 77–82. The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?

Key Concepts

Direct Proportionality

Direct proportionality means that two quantities increase or decrease in tandem. In this context, the distance fallen by a body is directly proportional to the square of the time of the fall, which can be expressed mathematically as d = k * t^2, where d is distance, t is time, and k is a constant of proportionality.

Quadratic Relationships

Quadratic relationships involve equations where the variable is raised to the second power. In this problem, since distance is proportional to the square of time, the relationship can be modeled by a quadratic equation, allowing us to predict distances for different time intervals by substituting values into the equation.

Unit Conversion and Scaling

Unit conversion and scaling are essential for solving problems involving different measurements. In this case, understanding how to scale the distance fallen from 3 seconds to 10 seconds involves using the proportionality constant derived from the initial condition (144 feet in 3 seconds) to find the new distance, ensuring consistent units throughout the calculation.