To describe the variation where $M$ varies directly with the square of $d$ and inversely with the square root of $x$, we can set up the following general formula:

$M = k \frac{d^2}{\sqrt{x}}$

Where:

• $k$ is a constant to be determined.
• $M$ varies directly with $d^2$ (the square of $d$) and inversely with $\sqrt{x}$ (the square root of $x$).

Now, using the given information that $M = 24$, $d = 3$, and $x = 25$, we can substitute these values into the equation to find $k$:

$24 = k \frac{3^2}{\sqrt{25}}$

Simplifying the equation:

$24 = k \frac{9}{5}$

Multiply both sides by 5:

$120 = 9k$

Solve for $k$:

$k = \frac{120}{9} = \frac{40}{3}$

Now, substitute the value of $k$ back into the general equation:

$M = \frac{40}{3} \frac{d^2}{\sqrt{x}}$

Thus, the formula that describes the variation is:

$M = \frac{40 d^2}{3 \sqrt{x}}$

Would you like more details or have any further questions?

Here are 5 questions that relate to this concept:

1. What is the value of $M$ when $d = 4$ and $x = 16$?
2. How does the formula change if $M$ varies directly with the cube of $d$ instead of the square?
3. Can you determine $d$ if $M = 50$ and $x = 36$?
4. What is the significance of the constant $k$ in direct and inverse variations?
5. How would the formula be affected if $M$ varied inversely with $d$?

Tip: When working with direct and inverse variations, always try to isolate the constant to understand how each variable affects the outcome.