To find the equation connecting [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( Z \)[/tex], we use the given conditions that [tex]\( x \)[/tex] is directly proportional to [tex]\( y \)[/tex] and inversely proportional to [tex]\( Z \)[/tex]. This relationship can be mathematically expressed as:
[tex]\[ x \propto \frac{y}{Z} \][/tex]
To convert this proportional relationship into an equation, we introduce a constant of proportionality [tex]\( k \)[/tex]:
[tex]\[ x = k \frac{y}{Z} \][/tex]
We are given that when [tex]\( x = 15 \)[/tex], [tex]\( y = 10 \)[/tex], and [tex]\( Z = 4 \)[/tex]. We can use these values to solve for the constant [tex]\( k \)[/tex].
Substituting the given values into the equation:
[tex]\[ 15 = k \frac{10}{4} \][/tex]
Simplify the fraction on the right-hand side:
[tex]\[ 15 = k \cdot 2.5 \][/tex]
To solve for [tex]\( k \)[/tex], divide both sides of the equation by 2.5:
[tex]\[ k = \frac{15}{2.5} = 6 \][/tex]
We have found that the constant of proportionality [tex]\( k \)[/tex] is 6. Using this constant, we can now write the equation connecting [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( Z \)[/tex]:
[tex]\[ x = 6 \frac{y}{Z} \][/tex]
Thus, the equation that connects [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( Z \)[/tex] is:
[tex]\[ x = 6 \frac{y}{Z} \][/tex]
