https://file.aiquickdraw.com/mathbot/file/172575881113532fmw4ee.jpg

The problem in the image states:

"P varies jointly as \( T \) and the square of \( Q \), and \( P = 39 \) when \( T = 18 \) and \( Q = 6 \). Find \( P \) when \( T = 4 \) and \( Q = 5 \)."

### Step 1: Write the equation for joint variation
Since \( P \) varies jointly as \( T \) and \( Q^2 \), the relationship can be expressed as:
\[
P = k \cdot T \cdot Q^2
\]
where \( k \) is the constant of proportionality.

### Step 2: Find the constant \( k \)
We are given that \( P = 39 \), \( T = 18 \), and \( Q = 6 \). Plug these values into the equation to solve for \( k \):
\[
39 = k \cdot 18 \cdot 6^2
\]
\[
39 = k \cdot 18 \cdot 36
\]
\[
39 = 648k
\]
\[
k = \frac{39}{648} = \frac{1}{16.615} = \frac{1}{16.615} \approx 0.0602
\]

### Step 3: Find \( P \) when \( T = 4 \) and \( Q = 5 \)
Now that we have the value of \( k \), substitute \( k \), \( T = 4 \), and \( Q = 5 \) into the equation to find \( P \):
\[
P = 0.0602 \times 4 \times 5^2
\]
\[
P = 0.0602 \times 4 \times 25
\]
\[
P = 0.0602 \times 100
\]
\[
P = 6.02
\]

### Final Answer
So, \( P \) is approximately **6.02** when \( T = 4 \) and \( Q = 5 \).

---

Would you like further details or have any questions?

### 5 Related Questions:
1. How do you determine the constant of proportionality in joint variation problems?
2. What is the difference between direct, inverse, and joint variation?
3. How would the problem change if \( P \) varied inversely with \( Q^2 \)?
4. Can joint variation include more than two variables? If so, how?
5. How would you solve for \( P \) if only \( T \) or \( Q \) were given, but not both?

### Tip:
Always double-check your calculations when dealing with constants of proportionality, as a small mistake can lead to an incorrect final answer.

P varies jointly as T and the square of Q, and P = 39 when T = 18 and Q = 6. Find P when T = 4 and Q = 5.

Solve the problem where P varies jointly as T and Q^2. Given P = 39 when T = 18 and Q = 6, find P when T = 4 and Q = 5. This example covers joint variation principles and algebraic concepts, suitable for Grades 9-12.

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