SOLUTION:
Step 1:
In this question, we are given the following:
Step 2:
The details of the solution are as follows:
PART A:
[tex]\begin{gathered} The\text{ mathematical model would be:} \\ F\text{ = }\frac{k\text{ m}_1m_2}{r^2},\text{ where k is a constant of proportionality} \end{gathered}[/tex]
PART B:
[tex]\begin{gathered} The\text{ mathematical model would be:} \\ F\text{ = k rs , where k is a constant of proportionality} \\ when\text{ F = 1344 , r = 12 , s = 2 , } \\ Then,\text{ we have that:} \\ 1344\text{ = k \lparen12\rparen \lparen2\rparen} \\ \text{24 k = 1344} \\ Divide\text{ both sides by 24, we have that:} \\ k\text{ =}\frac{1344}{24} \\ k\text{ = 56} \\ Hence,\text{ the mathematical model would now be:} \\ F\text{ = 56 r s} \end{gathered}[/tex]
PART C:
[tex]\begin{gathered} T\text{he mathematical model would be:} \\ z\text{ =}\frac{k\text{ x}^2}{y}\text{ , where k is a constant of proportionality} \\ where\text{ z =9, x =9 , y =6 . Then, we have that:} \\ 9\text{ =}\frac{k(9)^2}{6} \\ 9\text{ =}\frac{81k}{6} \\ cross-multiply,\text{ we have that:} \\ \text{9 x 6 = 81 k} \\ k\text{ =}\frac{54}{81} \\ k\text{ =}\frac{2}{3} \\ Hence,\text{ the mathematical model would be:} \\ z\text{ =}\frac{2}{3}\frac{x^2}{y} \end{gathered}[/tex]
