Since $p$ varies directly with $q$, we can express this relationship as:

$p = k \cdot q$

where $k$ is the constant of proportionality.

Step 1: Find $k$

Given $p = 144$ when $q = 8$, substitute these values into the equation:

$144 = k \cdot 8$

Solve for $k$:

$k = \frac{144}{8} = 18$

Thus, the constant of proportionality $k$ is 18.

Step 2: Function that Relates $p$ and $q$

Now we can write the function as:

$p = 18 \cdot q$

Step 3: Find $p$ when $q = 4$

Substitute $q = 4$ into the equation:

$p = 18 \cdot 4 = 72$

So, when $q = 4$, $p = 72$.

---

Summary

• Constant of proportionality $k = 18$.
• Function relating $p$ and $q$: $p = 18 \cdot q$.
• When $q = 4$, $p = 72$.

Would you like more details on any of these steps?

---

Related Questions

1. What is direct variation and how does it differ from inverse variation?
2. If $p$ varies inversely with $q$, how would we represent this relationship?
3. How would we determine $q$ if $p$ is given, using the equation $p = 18q$?
4. Can direct variation have a negative constant of proportionality?
5. How would the equation change if $p$ varied directly with the square of $q$?

---

Tip

In direct variation, if one variable increases, the other increases proportionally.