Question: The width of a rectangular tank is $x$ cm. Its height is $\frac{2}{3}$ its width and its length is four times its height.
The volume of the tank (in litres) is
1. **State the problem:**
We need to find the volume of a rectangular tank where:
- Width = $x$ cm
- Height = $\frac{2}{3}x$ cm
- Length = 4 times the height = $4 \times \frac{2}{3}x = \frac{8}{3}x$ cm
2. **Formula for volume of a rectangular tank:**
$$\text{Volume} = \text{length} \times \text{width} \times \text{height}$$
3. **Calculate the volume in cubic centimeters:**
$$V = \left(\frac{8}{3}x\right) \times x \times \left(\frac{2}{3}x\right) = \frac{8}{3}x \times x \times \frac{2}{3}x$$
4. **Simplify the expression:**
$$V = \frac{8}{3} \times \frac{2}{3} \times x \times x \times x = \frac{16}{9} x^3$$
5. **Convert cubic centimeters to litres:**
Since $1000$ cubic centimeters = 1 litre, we divide by $1000$:
$$V = \frac{16}{9} x^3 \times \frac{1}{1000} = \frac{16}{9} x^3 \times 10^{-3}$$
6. **Final answer:**
The volume of the tank in litres is:
$$\boxed{\frac{16}{9} x^3 \times 10^{-3} \text{ litres}}$$
This corresponds to option (a).