Write a general formula to describe the variation.
M varies directly with the square of d and inversely with the square root of​ x; Upper M equals 24 when d equals 3 and x equals 25

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.

Write a general formula to describe the variation. M varies directly with the square of d and inversely with the square root of​ x; M equals 24 when d equals 3 and x equals 25.

Explore how M varies directly with the square of d and inversely with the square root of x. This problem, designed for students in Grades 9-12, includes a step-by-step solution and the derived formula M = (40 * d^2) / (3 * sqrt(x)) with an example where M = 24, d = 3, and x = 25.

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