What is the volume of a compound solid made of a prism and a cone?

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The volume of a compound solid made of a prism and a cone is the sum of their individual volumes.

To find the volume of a compound solid consisting of a prism and a cone, you need to calculate the volume of each shape separately and then add them together. Let's break it down step by step.

First, for the prism, you need to know the area of its base and its height. The volume \( V_{\text{prism}} \) of a prism is given by the formula:
\[ V_{\text{prism}} = \text{Base Area} \times \text{Height} \]
For example, if the base of the prism is a rectangle with a length of 5 cm and a width of 3 cm, and the height of the prism is 10 cm, the base area would be \( 5 \times 3 = 15 \) square centimetres. Therefore, the volume of the prism would be \( 15 \times 10 = 150 \) cubic centimetres.

Next, for the cone, you need to know the radius of its base and its height. The volume \( V_{\text{cone}} \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height. For instance, if the radius of the cone's base is 2 cm and the height is 6 cm, the volume would be:
\[ V_{\text{cone}} = \frac{1}{3} \pi (2)^2 (6) = \frac{1}{3} \pi (4) (6) = \frac{1}{3} \pi (24) = 8\pi \approx 25.13 \text{ cubic centimetres} \]

Finally, add the volumes of the prism and the cone to get the total volume of the compound solid:
\[ V_{\text{total}} = V_{\text{prism}} + V_{\text{cone}} \]
Using our examples, this would be:
\[ V_{\text{total}} = 150 + 25.13 \approx 175.13 \text{ cubic centimetres} \]

So, by calculating the volumes of the individual shapes and adding them together, you can find the volume of the compound solid.