Understanding the Problem

We are given a conical vessel with a specific base radius and height. Water is poured into this vessel up to a certain fraction of its total capacity. We need to find the volume of the water inside the vessel.

Key Information Provided:

• Shape of the vessel: Cone
• Base Radius ($r$): 31 cm
• Height ($H$): 45 cm
• Water level: $\frac{2}{3}$ of the vessel's capacity

Formula for the Volume of a Cone

The volume ($V$) of a cone is calculated using the formula:

$$ V_{\text{cone}} = \frac{1}{3} \pi r^2 H $$

Where:

• $r$ is the radius of the base
• $H$ is the height of the cone
• $\pi$ is the mathematical constant pi

Calculating the Total Volume of the Conical Vessel

First, let's calculate the total volume of the conical vessel using the given radius and height.

Given $r = 31$ cm and $H = 45$ cm:

$$ V_{\text{vessel}} = \frac{1}{3} \pi (31 \text{ cm})^2 (45 \text{ cm}) $$

Calculate the square of the radius:

$$ (31 \text{ cm})^2 = 961 \text{ cm}^2 $$

Now substitute the values into the volume formula:

$$ V_{\text{vessel}} = \frac{1}{3} \pi (961 \text{ cm}^2) (45 \text{ cm}) $$

Simplify the calculation:

$$ V_{\text{vessel}} = \pi (961 \text{ cm}^2) \left(\frac{45}{3} \text{ cm}\right) $$

$$ V_{\text{vessel}} = \pi (961 \text{ cm}^2) (15 \text{ cm}) $$

$$ V_{\text{vessel}} = 14415 \pi \text{ cm}^3 $$

Calculating the Volume of Water

The problem states that water is poured into the vessel until it is $\frac{2}{3}$ full. To find the volume of water, we need to calculate $\frac{2}{3}$ of the total volume of the vessel.

Volume of water ($V_{\text{water}}$) = $\frac{2}{3} \times V_{\text{vessel}}$

$$ V_{\text{water}} = \frac{2}{3} \times 14415 \pi \text{ cm}^3 $$

Perform the multiplication:

$$ V_{\text{water}} = 2 \times \left(\frac{14415}{3}\right) \pi \text{ cm}^3 $$

Divide 14415 by 3:

$$ \frac{14415}{3} = 4805 $$

Now multiply by 2:

$$ V_{\text{water}} = 2 \times 4805 \pi \text{ cm}^3 $$

$$ V_{\text{water}} = 9610 \pi \text{ cm}^3 $$

Final Answer Check

The calculated volume of water in the vessel is $9610 \pi$ cm³. This value matches one of the given options.