Let's solve the problem step by step:
### Given:
- [tex]\( P_1 = 2 \)[/tex]
- [tex]\( Q_1 = 0.25 \)[/tex]
### Step 1: Determine the relationship between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]
Since [tex]\( Q \)[/tex] is inversely proportional to [tex]\( P \)[/tex], we have:
[tex]\[ Q \cdot P = k \][/tex]
where [tex]\( k \)[/tex] is a constant.
We can find [tex]\( k \)[/tex] using the given values of [tex]\( P_1 \)[/tex] and [tex]\( Q_1 \)[/tex]:
[tex]\[ k = Q_1 \cdot P_1 \][/tex]
[tex]\[ k = 0.25 \cdot 2 = 0.5 \][/tex]
### (i) Express [tex]\( Q \)[/tex] in terms of [tex]\( P \)[/tex]
Since [tex]\( Q \cdot P = k \)[/tex] and [tex]\( k = 0.5 \)[/tex]:
[tex]\[ Q = \frac{k}{P} = \frac{0.5}{P} \][/tex]
### (ii) Find the value of [tex]\( Q \)[/tex] when [tex]\( P = 5 \)[/tex]
Using the expression for [tex]\( Q \)[/tex] in terms of [tex]\( P \)[/tex]:
[tex]\[ Q = \frac{0.5}{5} \][/tex]
[tex]\[ Q = 0.1 \][/tex]
### (iii) Calculate the value of [tex]\( P \)[/tex] when [tex]\( Q = 0.2 \)[/tex]
We know that [tex]\( Q \cdot P = 0.5 \)[/tex]. If [tex]\( Q = 0.2 \)[/tex]:
[tex]\[ 0.2 \cdot P = 0.5 \][/tex]
[tex]\[ P = \frac{0.5}{0.2} \][/tex]
[tex]\[ P = 2.5 \][/tex]
### Summary of Results:
- The relationship between [tex]\( Q \)[/tex] and [tex]\( P \)[/tex] is: [tex]\( Q = \frac{0.5}{P} \)[/tex]
- [tex]\( Q = 0.1 \)[/tex] when [tex]\( P = 5 \)[/tex]
- [tex]\( P = 2.5 \)[/tex] when [tex]\( Q = 0.2 \)[/tex]
These are the step-by-step solutions to the given problems.
