$R$ is directly proportional to $L$ and inversely proportional to $P$. Given that $R = 3$ when $L = 9$ and $P = 0.8$, we want to find the value of $R$ when $L = 15$ and $P = 1.8$.

1. Problem Description

$R$ is directly proportional to $L$ and inversely proportional to $P$. Given that $R = 3$ when $L = 9$ and $P = 0.8$, we want to find the value of $R$ when $L = 15$ and $P = 1.8$.

2. Solution Steps

Since $R$ is directly proportional to $L$ and inversely proportional to $P$, we can write the relationship as:

$R = k \cdot \frac{L}{P}$, where $k$ is the constant of proportionality.

We are given that $R = 3$ when $L = 9$ and $P = 0.8$. We can use this information to find the value of $k$:

$3 = k \cdot \frac{9}{0.8}$

Multiply both sides by $0.8$:

$3 \cdot 0.8 = 9k$

$2.4 = 9k$

Divide both sides by $9$:

$k = \frac{2.4}{9} = \frac{24}{90} = \frac{4}{15}$

So, the relationship is:

$R = \frac{4}{15} \cdot \frac{L}{P}$

Now we want to find $R$ when $L = 15$ and $P = 1.8$. Substitute these values into the equation:

$R = \frac{4}{15} \cdot \frac{15}{1.8}$

$R = \frac{4}{1.8} = \frac{40}{18} = \frac{20}{9}$

$R = \frac{20}{9} = 2.222...$

Rounding to one decimal place, $R \approx 2.2$

3. Final Answer

The final answer is A. 2.2