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Please solve and show all the workings out

We are given that \( M \) is inversely proportional to \( g^3 \), meaning we can express this relationship as:

\[
M = \frac{k}{g^3}
\]

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

We are also given that when \( g = 2.5 \), \( M = 24 \). We can use this information to find \( k \).

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

Substitute \( M = 24 \) and \( g = 2.5 \) into the equation:

\[
24 = \frac{k}{(2.5)^3}
\]

First, calculate \( (2.5)^3 \):

\[
(2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625
\]

Now substitute this into the equation:

\[
24 = \frac{k}{15.625}
\]

Multiply both sides by 15.625 to solve for \( k \):

\[
k = 24 \times 15.625 = 375
\]

### Step 2: Write the formula for \( M \) in terms of \( g \)

Now that we have \( k = 375 \), substitute this value back into the original equation:

\[
M = \frac{375}{g^3}
\]

Thus, the formula for \( M \) in terms of \( g \) is:

\[
M = \frac{375}{g^3}
\]

This is the required formula.

---

Would you like more details on any specific step? 

Here are some related questions to further your understanding:

1. What happens to the value of \( M \) if \( g \) doubles?
2. If \( M = 3 \), what is the value of \( g \)?
3. Can you express \( g \) in terms of \( M \)?
4. How would the formula change if \( M \) was inversely proportional to \( g^2 \) instead of \( g^3 \)?
5. What is the physical meaning of inverse proportionality in real-world problems?

**Tip:** When solving inverse proportional problems, make sure you properly calculate and use the constant of proportionality, which relates the variables based on given conditions.

M is inversely proportional to g^3. M = 24 when g = 2.5. (a) Find a formula for M in terms of g.

Learn how to find the formula for M in terms of g when M is inversely proportional to g^3. Given M = 24 when g = 2.5, this problem demonstrates key concepts like inverse proportionality and exponentiation. Suitable for high school students.

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