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.