Question

there are 1000 litres in \\(1\text{ m}^3\\)

the water trough below is half of a cylinder.

what is the capacity of the water trough, to the nearest litre?

Explanation:

⚡ Using what you learned: volume of prisms, pyramids, cylinders, cones, spheres · Unit Conversions

Step 1: Identify the dimensions of the cylinder

The water trough is half of a horizontal cylinder.

• The depth (or radius \( r \)) of the semi-cylinder is \( 0.65\text{ m} \).
• The length (or height \( h \)) of the cylinder is \( 2.75\text{ m} \).

Step 2: Calculate the volume of the trough in cubic meters

The volume of a full cylinder is given by:

$$ V_{\text{cylinder}} = \pi r^2 h $$

Since the trough is half of a cylinder:

$$ V_{\text{trough}} = \frac{1}{2} \pi r^2 h $$

Substitute the given values:

$$ V_{\text{trough}} = \frac{1}{2} \times \pi \times (0.65)^2 \times 2.75 $$

$$ V_{\text{trough}} = \frac{1}{2} \times \pi \times 0.4225 \times 2.75 $$

$$ V_{\text{trough}} \approx 1.82479\text{ m}^3 $$

Step 3: Convert the volume to litres

Using the conversion factor \( 1\text{ m}^3 = 1000\text{ litres} \):

$$ \text{Capacity} = 1.82479 \times 1000 \approx 1824.79\text{ litres} $$

Rounding to the nearest litre:

$$ \text{Capacity} \approx 1825\text{ litres} $$

Answer:

1825 litres