$M$ 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 $M=224$.

Answer

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Answer

Solve for $r$: $r = \boxed{8}$

Steps

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 $r$: $r = \boxed{8}$