The problem states that the amount of material $M$ varies directly with $x$ and inversely with $y^3$. This can be expressed mathematically as:

$M = k \frac{x}{y^3}$

where $k$ is a constant of proportionality.

Step 1: Find the constant $k$

We are given that $M = 25$ when $x = 1$ and $y = 2$. Plugging these values into the equation:

$25 = k \frac{1}{2^3}$

Simplifying the equation:

$25 = k \frac{1}{8}$

Multiplying both sides by 8 to solve for $k$:

$k = 25 \times 8 = 200$

Step 2: Use the formula to find $M$ when $x = 15$ and $y = 3$

Now that we know $k = 200$, we can substitute $x = 15$ and $y = 3$ into the equation to find $M$:

$M = 200 \frac{15}{3^3}$

Simplifying:

$M = 200 \frac{15}{27} = 200 \times \frac{5}{9} = \frac{1000}{9}$

Thus, the value of $M$ is $\frac{1000}{9}$, which is approximately $111.11$.

None of the given choices seem to match this, so we may need to reconsider or recheck the question and choices. Could you let me know what you would like to do next?

Would you like a more detailed breakdown or help with a specific part of the question? Here are some follow-up questions you might be interested in:

1. How do we determine when a relationship is direct or inverse?
2. What is the general approach to finding constants in variation problems?
3. Why did we raise $y$ to the third power in this equation?
4. Could we approach this type of problem using another method?
5. Can you show more examples where quantities vary directly and inversely?

Tip: Always double-check your calculations and make sure you're aligning with the exact format of the answer choices.