In the following exercises, solve. The distance an object...

In the following exercises, solve.
The distance an object falls varies directly to the square of the time it falls. A ball falls 45 feet in 3 seconds.
(a) Write the equation that relates the distance to the time.
(b) How far will the ball fall in 7 seconds?

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Key Concepts

Direct Variation

Direct variation describes a relationship between two variables where the change in one variable is proportionally linked to the change in the other. This relationship can be expressed in the form y = kx, where k is the constant of proportionality. It allows one to compute unknown values by scaling one variable in direct accordance with the other.

Quadratic Relationships

A quadratic relationship in direct variation occurs when one variable changes in proportion to the square of another variable. This is typically expressed as y = kx². Such relationships are common in physics contexts, for instance, when describing the distance traveled under constant acceleration, where the distance is proportional to the square of the time elapsed.

Determining the Constant of Proportionality

In problems involving direct and quadratic relationships, determining the constant of proportionality is a crucial step. It involves substituting known pairs of variable values into the proportional relationship equation to solve for the constant. With this constant, one can then predict or calculate other values within the same proportional framework.

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