Given the information, we need to determine which equation correctly models the relationship between [tex]\( x \)[/tex], [tex]\( p \)[/tex], [tex]\( m \)[/tex], and [tex]\( y \)[/tex].
First, let's summarize the information provided:
- [tex]\( y = 4 \)[/tex]
- [tex]\( p = 0.5 \)[/tex]
- [tex]\( m = 2 \)[/tex]
- [tex]\( x = 2 \)[/tex]
It is stated that [tex]\( x \)[/tex] varies directly with the product of [tex]\( p \)[/tex] and [tex]\( m \)[/tex] and inversely with [tex]\( y \)[/tex]. Therefore, [tex]\( x \)[/tex] must follow the form:
[tex]\[ x = k \frac{p \cdot m}{y} \][/tex]
where [tex]\( k \)[/tex] is a constant.
Step 1: Determine the constant [tex]\( k \)[/tex]
Given the values:
[tex]\[ x = 2 \][/tex]
[tex]\[ p = 0.5 \][/tex]
[tex]\[ m = 2 \][/tex]
[tex]\[ y = 4 \][/tex]
Substitute these values into the equation:
[tex]\[ 2 = k \frac{0.5 \cdot 2}{4} \][/tex]
[tex]\[ 2 = k \frac{1}{4} \][/tex]
[tex]\[ 2 = \frac{k}{4} \][/tex]
[tex]\[ k = 2 \times 4 = 8 \][/tex]
So, the constant [tex]\( k \)[/tex] is 8.
Step 2: Verify which given equation is correct
1. [tex]\( x p r r y = 8 \)[/tex]
This equation is somewhat ambiguous, so moving on for now.
2. [tex]\[ \frac{x y}{p m} = 8 \][/tex]
Substitute the values to check:
[tex]\[ \frac{2 \cdot 4}{0.5 \cdot 2} = 8 \][/tex]
[tex]\[ \frac{8}{1} = 8 \][/tex]
This is true.
3. [tex]\[ \frac{x p m}{y} = 0.5 \][/tex]
Substitute the values to check:
[tex]\[ \frac{2 \cdot 0.5 \cdot 2}{4} = 0.5 \][/tex]
[tex]\[ \frac{2}{4} = 0.5 \][/tex]
This is also true.
4. [tex]\( \frac{x}{\rho m y}=0.5 \)[/tex]
This equation contains an undefined variable [tex]\( \rho \)[/tex], which we cannot verify without additional information.
Based on the substitutions:
Conclusion:
The verified equations that hold true are:
[tex]\[ \frac{x y}{p m} = 8 \][/tex]
[tex]\[ \frac{x p m}{y} = 0.5 \][/tex]
However, among these equations and given the context, it is necessary to identify a single model. Since both equations accurately describe the scenario and assuming context would provide priority, both could fit. But, typically, simpler equations are easier to interpret in a given context.
Therefore, the simpler and clear modeled equation from choices is:
[tex]\[ \frac{x y}{p m} = 8 \][/tex]
