Sure, let's work through the problem step by step.
Firstly, we are given the relationship [tex]\( P \propto QR^2 \)[/tex], which implies that:
[tex]\[ P = k \cdot Q \cdot R^2 \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
### Step 1: Determine the constant [tex]\( k \)[/tex]
Given the initial conditions:
[tex]\[ Q = 9, \ R = 6, \ P = 13.5 \][/tex]
We can plug these values into the equation to find [tex]\( k \)[/tex]:
[tex]\[ 13.5 = k \cdot 9 \cdot 6^2 \][/tex]
[tex]\[ 13.5 = k \cdot 9 \cdot 36 \][/tex]
[tex]\[ 13.5 = k \cdot 324 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{13.5}{324} \][/tex]
[tex]\[ k = 0.041666666666666664 \][/tex]
### Step 2: Find [tex]\( P \)[/tex] when [tex]\( Q = 12 \)[/tex] and [tex]\( R = 15 \)[/tex]
Using the found value of [tex]\( k \)[/tex]:
[tex]\[ k = 0.041666666666666664 \][/tex]
And the given values:
[tex]\[ Q = 12, \ R = 15 \][/tex]
We can substitute these into our original proportionality equation to find [tex]\( P \)[/tex]:
[tex]\[ P = k \cdot Q \cdot R^2 \][/tex]
[tex]\[ P = 0.041666666666666664 \cdot 12 \cdot 15^2 \][/tex]
[tex]\[ P = 0.041666666666666664 \cdot 12 \cdot 225 \][/tex]
[tex]\[ P = 112.5 \][/tex]
### Step 3: Find [tex]\( R \)[/tex] when [tex]\( P = 43 \)[/tex] and [tex]\( Q = 10.5 \)[/tex]
Using [tex]\( k \)[/tex] again:
[tex]\[ k = 0.041666666666666664 \][/tex]
And the given values:
[tex]\[ P = 43, \ Q = 10.5 \][/tex]
We need to find [tex]\( R \)[/tex], so we use:
[tex]\[ 43 = 0.041666666666666664 \cdot 10.5 \cdot R^2 \][/tex]
Solving for [tex]\( R^2 \)[/tex]:
[tex]\[ R^2 = \frac{43}{0.041666666666666664 \cdot 10.5} \][/tex]
[tex]\[ R^2 \approx 98.29490445859873 \][/tex]
Taking the square root to find [tex]\( R \)[/tex]:
[tex]\[ R \approx \sqrt{98.29490445859873} \][/tex]
[tex]\[ R \approx 9.913915184512842 \][/tex]
### Step 4: Determine the factor by which [tex]\( P \)[/tex] is multiplied when [tex]\( R \)[/tex] is multiplied by 3 and [tex]\( Q \)[/tex] is divided by 3
Given the transformations:
[tex]\[ R \rightarrow 3R \][/tex]
[tex]\[ Q \rightarrow \frac{Q}{3} \][/tex]
We need to see how [tex]\( P \)[/tex] changes under these conditions. Using the relationship [tex]\( P \propto QR^2 \)[/tex]:
[tex]\[ P_{\text{new}} \propto \left( \frac{Q}{3} \right) \left( 3R \right)^2 \][/tex]
This simplifies to:
[tex]\[ P_{\text{new}} \propto \left( \frac{Q}{3} \right) \cdot 9R^2 \][/tex]
[tex]\[ P_{\text{new}} \propto 3QR^2 \][/tex]
Thus, [tex]\( P \)[/tex] is multiplied by a factor of 3.
### Summary of Results:
1. The constant of proportionality, [tex]\( k \)[/tex], is approximately [tex]\( 0.041666666666666664 \)[/tex].
2. When [tex]\( Q = 12 \)[/tex] and [tex]\( R = 15 \)[/tex], [tex]\( P \)[/tex] is [tex]\( 112.5 \)[/tex].
3. When [tex]\( P = 43 \)[/tex] and [tex]\( Q = 10.5 \)[/tex], [tex]\( R \)[/tex] is approximately [tex]\( 9.913915184512842 \)[/tex].
4. If [tex]\( R \)[/tex] is multiplied by 3 and [tex]\( Q \)[/tex] is divided by 3, [tex]\( P \)[/tex] is multiplied by a factor of 3.
