To determine how many hours Natasha was asleep on the beach, we first need to calculate $\dfrac{3}{4}$ of the total time she spent on the beach.

The total time Natasha spent on the beach was $1 \dfrac{1}{2}$ hours, which can be written as a mixed number or an improper fraction:

$1 \dfrac{1}{2} = \dfrac{3}{2} \text{ hours}$

Next, we calculate $\dfrac{3}{4}$ of this time:

$\text{Time asleep} = \dfrac{3}{4} \times \dfrac{3}{2}$

To multiply these fractions, multiply the numerators and the denominators:

$\text{Time asleep} = \dfrac{3 \times 3}{4 \times 2} = \dfrac{9}{8}$

Now, $\dfrac{9}{8}$ is an improper fraction, which can be converted to a mixed number:

$\dfrac{9}{8} = 1 \dfrac{1}{8} \text{ hours}$

Thus, Natasha was asleep for 1 $\dfrac{1}{8}$ hours on the beach.

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Would you like more details or have any questions about this?

Here are 5 related questions to explore:

1. What is $\dfrac{3}{4}$ of a different time, like 2 hours?
2. How do you convert a mixed number to an improper fraction?
3. How do you multiply fractions?
4. How do you add fractions with different denominators?
5. What happens if Natasha spent less time on the beach—how would that affect the time she was asleep?

Tip: When dealing with fractions of mixed numbers, always convert the mixed number to an improper fraction first before performing operations.