p
  varies directly with  
q
  and when  
q
=
8
 ,  
p
=
144
 


Give the constant of proportionality,  
k
=
  
  

Give the function that relates the relates p and q. 
  

If  
q
=
4
 , then  
p
=
 


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.

p varies directly with q and when q = 8, p = 144. Give the constant of proportionality, k. Give the function that relates p and q. If q = 4, then p = ?

Understand how to find the constant of proportionality, k, in a direct variation problem where p varies directly with q. Given values q = 8 and p = 144, solve for k, form the function relating p and q, and find p when q = 4. Ideal for Grades 6-8.

Math Problem Statement

p varies directly with
q and when
q

8 ,
p

Give the constant of proportionality,
k

If
q

4 , then
p

Solution

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