Calculate the semi-perimeter s.

Here are the solutions to the problems.

1. Find the area of the triangle with sides $8$m, $9$m, and $15$m.

Step 1: Calculate the semi-perimeter $s$.
The semi-perimeter $s$ is half the sum of the side lengths $a, b, c$.

$$s = \frac{a+b+c}{2} = \frac{8+9+15}{2} = \frac{32}{2} = 16 \text{ m}$$

Step 2: Apply Heron's formula for the area of a triangle.
The area $A$ is given by $A = \sqrt{s(s-a)(s-b)(s-c)}$.

$$A = \sqrt{16(16-8)(16-9)(16-15)}$$ $$A = \sqrt{16 \times 8 \times 7 \times 1}$$ $$A = \sqrt{896}$$ $$A \approx 29.933 \text{ m}^2$$

Rounding to one decimal place:
The area of the triangle is $\boxed{29.9 \text{ m}^2}$.

2. Find the area of the parallelogram with sides $9$cm and $12$cm, and an angle of $48^\circ$ between them.

Step 1: Use the formula for the area of a parallelogram.
The area $A$ of a parallelogram with sides $a$ and $b$ and angle $\theta$ between them is $A = ab \sin \theta$.

$$A = 9 \text{ cm} \times 12 \text{ cm} \times \sin 48^\circ$$ $$A = 108 \times \sin 48^\circ$$

Step 2: Calculate the value.
Using $\sin 48^\circ \approx 0.74314$:

$$A = 108 \times 0.74314$$ $$A \approx 80.259 \text{ cm}^2$$

Rounding to two decimal places:
The area of the parallelogram is $\boxed{80.26 \text{ cm}^2}$.

3. Find the area of triangle PQR with $PQ = 7.5$, $PR = 8.3$, and $\angle P = 56^\circ$.

Step 1: Use the formula for the area of a triangle given two sides and the included angle.
The area $A$ is given by $A = \frac{1}{2} PQ \times PR \times \sin P$.

$$A = \frac{1}{2} \times 7.5 \times 8.3 \times \sin 56^\circ$$

Step 2: Calculate the value.

$$A = \frac{1}{2} \times 62.25 \times \sin 56^\circ$$

Using $\sin 56^\circ \approx 0.82904$:

$$A = 31.125 \times 0.82904$$ $$A \approx 25.804$$

Rounding to two decimal places:
The area of triangle PQR is $\boxed{25.80}$.

4. Calculate the capacity in litres of a container that measures $2$m by $1.5$m by $1.2$m.

Step 1: Calculate the volume of the container in cubic meters.
The volume $V$ of a rectangular container is $V = \text{length} \times \text{width} \times \text{height}$.

$$V = 2 \text{ m} \times 1.5 \text{ m} \times 1.2 \text{ m}$$ $$V = 3.6 \text{ m}^3$$

Step 2: Convert the volume from cubic meters to litres.
Since $1 \text{ m}^3 = 1000 \text{ litres}$:

$$\text{Capacity} = 3.6 \text{ m}^3 \times 1000 \text{ litres/m}^3$$ $$\text{Capacity} = 3600 \text{ litres}$$

The capacity of the container is $\boxed{3600 \text{ litres}}$.

5. Solve the ratio problems.