To determine which equation correctly represents [tex]\( F \)[/tex], the force on an object due to gravity according to [tex]\( m \)[/tex], the object's mass, let's examine the given information step by step:
1. Direct Variation of Force with Mass:
- The problem states that the force of gravity on an object varies directly with its mass. This implies that as the mass [tex]\( m \)[/tex] increases or decreases, the force [tex]\( F \)[/tex] increases or decreases proportionally. Mathematically, if two quantities vary directly, they can be expressed as [tex]\( F = k \cdot m \)[/tex], where [tex]\( k \)[/tex] is the constant of variation.
2. Constant of Variation:
- The problem provides that the constant of variation due to gravity is [tex]\( 32.2 \)[/tex] feet per second squared. This means [tex]\( k = 32.2 \)[/tex].
3. Formulating the Equation:
- Plugging the constant of variation [tex]\( 32.2 \)[/tex] into the equation of direct variation, we get:
[tex]\[
F = 32.2 \cdot m
\][/tex]
4. Choosing the Correct Equation:
- The correct equation must match the form [tex]\( F = 32.2 \cdot m \)[/tex].
- Among the provided options:
- [tex]\( F = 16.1 m \)[/tex] is incorrect since the constant is not [tex]\( 16.1 \)[/tex].
- [tex]\( F = \frac{16.1}{m^2} \)[/tex] is incorrect as it represents inverse square variation.
- [tex]\( F = 32.2 m \)[/tex] is correct as it matches our derived equation.
- [tex]\( F = \frac{32.2}{m^2} \)[/tex] is incorrect for the same reason as above.
Thus, the equation that represents [tex]\( F \)[/tex], the force on an object due to gravity according to [tex]\( m \)[/tex], the object's mass, is:
[tex]\[ \boxed{F = 32.2 m} \][/tex]
