We are told that the amount of medicine (m) is proportional to the patient's body weight (w). Hence:
[tex]m\text{ }\propto\text{ w}[/tex]
We can re-write this as:
[tex]\begin{gathered} m\text{ = kw} \\ \text{where k is the constant of proportionality} \end{gathered}[/tex]
Given:
A patient weighing 116 pounds requires 126 milligrams.
Substituting the given data:
[tex]\begin{gathered} 126\text{ = k }\times\text{ 116} \\ k\text{ = }\frac{126}{116} \\ k\text{ = }\frac{63}{58} \end{gathered}[/tex]
The equation relating m and w is thus:
[tex]m\text{ = }\frac{63}{58}w[/tex]
The amount of medicine for a patient weighing 174 pounds can be calculated by substitution:
[tex]\begin{gathered} m\text{ = }\frac{63}{58}\text{ }\times\text{ 174} \\ m\text{ = 189} \end{gathered}[/tex]
Hence, the amount of medicine is 189 milligrams
