$M$ 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 $r$ : $r = 8$

Steps

Step 1 ： $M = k r^{2}$

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 $M$ : $M = 1008$

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 $r$ : $r = 8$