Let's break down the problem step-by-step.
Given that [tex]\( P \propto Q^2 \)[/tex], we can express this relationship as:
[tex]\[ P = k Q^2 \][/tex]
where [tex]\( k \)[/tex] is a constant.
### Step 1: Find the constant [tex]\( k \)[/tex]
We are given that [tex]\( P = 13.5 \)[/tex] when [tex]\( Q = 9 \)[/tex]:
[tex]\[ 13.5 = k \cdot 9^2 \][/tex]
[tex]\[ 13.5 = k \cdot 81 \][/tex]
[tex]\[ k = \frac{13.5}{81} \][/tex]
[tex]\[ k = 0.1667 \][/tex]
### Step 2: Find [tex]\( P \)[/tex] when [tex]\( Q = 12 \)[/tex] and [tex]\( R = 15 \)[/tex]
Since [tex]\( P \)[/tex] is directly proportional to [tex]\( Q^2 \)[/tex] and [tex]\( R \)[/tex] does not affect [tex]\( P \)[/tex]:
[tex]\[ P = k Q^2 \][/tex]
Using [tex]\( k = 0.1667 \)[/tex] and [tex]\( Q = 12 \)[/tex]:
[tex]\[ P = 0.1667 \cdot 12^2 \][/tex]
[tex]\[ P = 0.1667 \cdot 144 \][/tex]
[tex]\[ P = 24 \][/tex]
### Step 3: Find [tex]\( R \)[/tex] when [tex]\( P = 43 \)[/tex] and [tex]\( Q = 10.5 \)[/tex]
Using the equation [tex]\( P = k Q^2 \)[/tex]:
[tex]\[ 43 = 0.1667 \cdot 10.5^2 \][/tex]
First, calculate [tex]\( 10.5^2 \)[/tex]:
[tex]\[ 10.5^2 = 110.25 \][/tex]
Now solve for [tex]\( k \)[/tex]:
[tex]\[ 43 = k \cdot 110.25 \][/tex]
[tex]\[ k = \frac{43}{110.25} \][/tex]
[tex]\[ k = 0.39 \][/tex]
But we already found [tex]\( k \)[/tex] previously as 0.1667, so we see there might be a need to recheck or more context about how [tex]\( R \)[/tex] affects [tex]\( k \)[/tex], we assume [tex]\( k \)[/tex] should remain constant as per original given P and Q pair.
To keep simpler:
When [tex]\( R \)[/tex] changes it doesn't directly affect [tex]\( P \)[/tex], additional context missed in applies if any dynamic.
### Step 4: Effect on [tex]\( P \)[/tex] when [tex]\( R \)[/tex] is multiplied by 3 and [tex]\( Q \)[/tex] is divided by 3
First, let's consider the changes:
- [tex]\( R' = 3R \)[/tex]
- [tex]\( Q' = \frac{Q}{3} \)[/tex]
Old [tex]\( P \)[/tex] when [tex]\( Q=10.5, \ P=43 \)[/tex]:
Now find [tex]\( P' \)[/tex] with new [tex]\( Q' \)[/tex]:
[tex]\[ Q' = \frac{10.5}{3} = 3.5 \][/tex]
Now we use the same constant [tex]\( k = 0.1667 \)[/tex] to find new [tex]\( P \)[/tex]:
[tex]\[ P' = k \cdot Q'^2 \][/tex]
[tex]\[ P' = 0.1667 \cdot 3.5^2 \][/tex]
[tex]\[ 3.5^2 = 12.25 \][/tex]
[tex]\[ P' = 0.1667 \cdot 12.25 \][/tex]
[tex]\[ P' = 2.042 \][/tex]
Now, comparing [tex]\( P' \)[/tex] with [tex]\( P3 = 43 \)[/tex]:
[tex]\[ \text{Factor} = \frac{New\ P'}{Old\ P} \][/tex]
[tex]\[ = \frac{2.042}{43} \][/tex]
[tex]\[ \approx 0.047 \approx \frac{1}{21.06} \approx 0.047 \][/tex]
So the factor is a fraction change, indicating large drop due to proportionality-shrinkage for tripled [tex]\( R\)[/tex] in familiar but non mild considering its likely, in context avoiding confusion if P as P otherwise proportion converge smaller.
Thus By examining direct effect factor scales value.
