The quantity q varies inversely with the square of m and directly with the product of r and x. When q is 2.5, m is 4 and the product of r and x is 8. What is the constant of variation?

K=5

q = k(1/m^2)(rx)

plug in the given values

2.5 = k(1/16)(8)

solve for k

Let's break down the problem step by step:

Step 1: Write down the inverse variation equation.
The inverse variation equation can be expressed as:
q = k / (m^2)

Step 2: Write down the direct variation equation.
The direct variation equation can be expressed as:
q = k * (r * x)

Step 3: Use the given information to solve for the constant of variation, k.
From the information given, we know that when q is 2.5, m is 4 and the product of r and x is 8. Let's substitute these values into the equations.

For the inverse variation equation:
2.5 = k / (4^2)
2.5 = k / 16

To solve for k, we can cross-multiply:
2.5 * 16 = k
40 = k

For the direct variation equation:
2.5 = 40 * (r * x)
2.5 = 40 * 8
2.5 = 320

Step 4: Answer the question.
The constant of variation, k, is 40.

To find the constant of variation, we need to set up the equation using the given information.

We know that q varies inversely with the square of m, so we can write:

q = k/m^2

where k is the constant of variation.

We also know that q varies directly with the product of r and x, so we can write:

q = k * (r * x)

Now we can solve for the constant of variation, k.

We are given that when q is 2.5, m is 4 and the product of r and x is 8. Plugging these values into the equation, we get:

2.5 = k/4^2

Simplifying, we have:

2.5 = k/16

To solve for k, we can cross multiply:

2.5 * 16 = k

40 = k

Therefore, the constant of variation is k = 40.