If p varies as the cube of and q varies directly as the the square of r what is the relationship between paund r

The only way I can make sense of what you wrote is to interpret it as

p = mq^3
q = nr^2

p = m(nr^2)^3 = mn^3 r^6 = kr^k

p varies directly as the 6th power of r.

To determine the relationship between p and r, we need to combine the information given: "p varies as the cube of q" and "q varies directly as the square of r."

Let's break this down step by step:

1. "p varies as the cube of q": This means that p can be expressed as p = k(q^3), where k is a constant of variation. This implies that when q increases or decreases, p will change accordingly.

2. "q varies directly as the square of r": This implies that q can be expressed as q = kr^2, where k is another constant of variation. This means that when r increases or decreases, q will change proportionally.

Now, we can substitute the expression for q into the equation for p to find the relationship between p and r:

p = k(q^3)
p = k((kr^2)^3) (substituting q = kr^2)
p = k(k^3r^6)
p = k^4r^6

Therefore, the relationship between p and r is p varies as the sixth power of r (p ∝ r^6).