The quantity $$ q $$ varies inversely with the square of $$ m $$ and directly with the product of $$ r $$ and $$ x $$. When $$ m $$ is 4 and the product of $$ r $$ and $$ x $$ is $$ 8, q $$ is 2.5 . What is the constant of variation? $$ \frac{5}{8} $$ $$ \frac{5}{4} $$ $$ \frac{5}{10} $$

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To solve the problem, we start by expressing the relationship described:

1. Since $$ q $$ varies inversely with the square of $$ m $$ and directly with the product of $$ r $$ and $$ x $$, we can write the equation as:
   $$
   q = k \cdot \frac{r \cdot x}{m^2}
   $$
   where $$ k $$ is the constant of variation.

2. We know that when $$ m = 4 $$, $$ r \cdot x = 8 $$, and $$ q = 2.5 $$. We can substitute these values into the equation to find $$ k $$:
   $$
   2.5 = k \cdot \frac{8}{4^2}
   $$

3. Now, we calculate $$ 4^2 $$:
   $$
   4^2 = 16
   $$

4. Substitute this back into the equation:
   $$
   2.5 = k \cdot \frac{8}{16}
   $$

5. Simplifying $$ \frac{8}{16} $$:
   $$
   \frac{8}{16} = \frac{1}{2}
   $$

6. Now, we have:
   $$
   2.5 = k \cdot \frac{1}{2}
   $$

7. To find $$ k $$, multiply both sides by 2:
   $$
   k = 2.5 \cdot 2
   $$

8. Calculate $$ k $$:
   $$
   k = 5
   $$

Now, we have determined that the constant of variation $$ k $$ is 5.

Since the options provided are $$ \frac{5}{8} $$, $$ \frac{5}{4} $$, and $$ \frac{5}{10} $$, we can express $$ k $$ in terms of these fractions:

- $$ \frac{5}{8} = 0.625 $$
- $$ \frac{5}{4} = 1.25 $$
- $$ \frac{5}{10} = 0.5 $$

None of these options match $$ k = 5 $$. Therefore, it seems there might be a misunderstanding in the options provided. The correct constant of variation is $$ 5 $$.
The constant of variation is 5.

The quantity varies inversely with the square of and directly with the product of and . When is 4 and the product
of and is is 2.5 . What is the constant of variation?

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The quantity varies inversely with the square of and directly with the product of and . When is 4 and the product of and is is 2.5 . What is the constant of variation?