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The problem states that \( M \) is directly proportional to \( r^3 \), which can be expressed mathematically as:

\[
M = k \cdot r^3
\]

where \( k \) is the constant of proportionality.

### Step 1: Finding the constant \( k \)

From the given information, when \( r = 4 \), \( M = 160 \). Using this to find \( k \):

\[
160 = k \cdot 4^3
\]
\[
160 = k \cdot 64
\]
\[
k = \frac{160}{64} = 2.5
\]

Now we know that the equation for \( M \) is:

\[
M = 2.5 \cdot r^3
\]

### Part (a): Find \( M \) when \( r = 2 \)

Using the equation \( M = 2.5 \cdot r^3 \), substitute \( r = 2 \):

\[
M = 2.5 \cdot 2^3
\]
\[
M = 2.5 \cdot 8 = 20
\]

So, when \( r = 2 \), \( M = 20 \).

### Part (b): Find \( r \) when \( M = 540 \)

Using the equation \( M = 2.5 \cdot r^3 \), substitute \( M = 540 \) and solve for \( r \):

\[
540 = 2.5 \cdot r^3
\]
\[
r^3 = \frac{540}{2.5} = 216
\]
\[
r = \sqrt[3]{216} = 6
\]

So, when \( M = 540 \), \( r = 6 \).

### Final Answers:
- (a) \( M = 20 \) when \( r = 2 \)
- (b) \( r = 6 \) when \( M = 540 \)

Would you like further details or explanations on any of the steps?

Here are some related questions you might explore:
1. What if \( M \) were inversely proportional to \( r^3 \)?
2. How would the calculations change if \( M \) were proportional to \( r^2 \) instead of \( r^3 \)?
3. Can you graph the relationship between \( M \) and \( r \) given the proportionality?
4. What if \( M = 1000 \)? What would \( r \) be?
5. How can you verify the cube root of a large number like 216 manually?

**Tip**: For directly proportional relationships, the ratio \( \frac{M}{r^3} \) remains constant. This is a useful check when solving similar problems.

M is directly proportional to r^3. When r = 4, M = 160. a) Work out the value of M when r = 2. b) Work out the value of r when M = 540.

Learn how to solve problems involving direct proportionality where M is directly proportional to r^3. This question covers finding M when r = 2 and solving for r when M = 540. Ideal for students in Grades 9-11.

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