- Understand that direct variation between two variables \(F\) and \(m\) is defined by the formula \(F = k \cdot m\), where \(k\) is the constant of variation.
- Identify the constant of variation given in the problem as \(k = 32.2\).
- Substitute the value of \(k\) into the formula to obtain the equation \(F = 32.2 m\).
- Match this equation to the correct provided option: \(\boxed{F = 32.2 m}\).
Explanation
- Understanding Direct Variation
The problem states that the force of gravity \(F\) varies directly with the mass \(m\) of an object. In mathematics, direct variation between two variables \(F\) and \(m\) means that their relationship can be expressed by the equation \(F = k \cdot m\), where \(k\) is a constant known as the constant of variation. We are given that this constant of variation due to gravity is \(32.2\) feet per second squared. Therefore, we set \(k = 32.2\).
- Formulating the Equation
With the relationship \(F = k \cdot m\) and the constant \(k = 32.2\), we can substitute the value of \(k\) directly into the equation. This yields the final equation for the force of gravity \(F\) as a function of mass \(m\):
\[F = 32.2 m\]
Comparing this result to the given options, it is clear that \(F = 32.2 m\) is the correct representation.
Examples
In everyday life, direct variation is common. For example, if you are buying apples at a fixed price per pound, the total cost varies directly with the weight of the apples. If the price is \(\$2\) per pound, the total cost \(C\) in dollars for a weight \(w\) in pounds is given by \(C = 2w\). Just like the gravity problem, knowing the constant of variation allows you to predict the outcome for any given value of the independent variable.