$M$ is directly proportional to $r^{2}$ When $r=2, M=14$ a) Work out the value of $M$ when $r=12$. b) Work out the value of $r$ when $M=224$.

Step 1 ：\(M = kr^2\)

Step 2 ：Plug in \(r=2\) and \(M=14\): \(14 = k(2)^2\)

Step 3 ：Solve for \(k\): \(k = \frac{14}{4} = \frac{7}{2}\)

Step 4 ：a) Plug in \(r=12\) and \(k=\frac{7}{2}\): \(M = \frac{7}{2}(12)^2\)

Step 5 ：Calculate \(M\): \(M = \boxed{1008}\)

Step 6 ：b) Plug in \(M=224\) and \(k=\frac{7}{2}\): \(224 = \frac{7}{2}r^2\)

Step 7 ：Solve for \(r^2\): \(r^2 = \frac{2 \times 224}{7} = 64\)

Step 8 ：Solve for \(r\): \(r = \boxed{8}\)