It is given that a solid is composed of a cube with a side length of 6 meters and a hemisphere with a diameter of 6 meters. Let's use this information to make a diagram.
To find the volume of the solid, we need to find the volumes of the cube and the hemisphere and then add them. Let's deal with one thing at a time.
Volume of the Cube
The with a side length of a is calculated by the following formula.
\begin{aligned}
V_\text{cube}=a^3
\end{aligned}We know that the side length of the cube is 6 m. Let's substitute 6 for a into the formula and calculate S_\text{cube}.
V_\text{cube}=a^3
V_\text{cube}={\color{#0000FF}{6}}^3
V_\text{cube}=216
Therefore, the volume of our cube is 216 cubic meters.
Volume of the Hemisphere
Let's recall that the with a radius of r is determined by the following formula.
\begin{aligned}
V_\text{sphere}=\dfrac{4}{3}\pi r^3
\end{aligned}
A hemisphere is half the sphere, so its volume is half the volume of the sphere.
\begin{aligned}
V_\text{hemisphere}=\dfrac{\dfrac{4}{3}\pi r^3}{2}=\dfrac{4}{6}\pi r^3
\end{aligned}
We are told that the of the hemisphere is 6 meters. Dividing this value by 2, we get that its is 3 meters. Let's substitute 3 for r into the formula and find V_\text{hemisphere}.
V_\text{hemisphere}=\dfrac{4}{6}\pi r^3
V_\text{hemisphere}=\dfrac{4}{6}\pi ({\color{#0000FF}{3}})^3
V_\text{hemisphere}=\dfrac{4}{6}\pi \cdot 27
V_\text{hemisphere}=56.548667\ldots
V_\text{hemisphere}\approx 56.55
The volume of the hemisphere is approximately 56.5 cubic meters.
Volume of the Solid
Let's gather the information that we have found.
\begin{aligned}
V_\text{cube}&=216 \text{ m}^3\\
V_\text{hemisphere}&\approx 56.55 \text{ m}^3
\end{aligned}
By adding these volumes, we can find the volume of the whole solid.
\begin{aligned}
V_\text{solid}=216+56.55=272.55\text{ m}^3
\end{aligned}
Therefore, the volume of the composite solid is about 272.55m^3.
