Question
A solid is in the shape of a cone standing on a hemisphere with both
their radii being equal to 1 cm and the height of the cone is equal to
its radius. Find the volume of the solid in terms of π.
their radii being equal to 1 cm and the height of the cone is equal to
its radius. Find the volume of the solid in terms of π.
Answers
GPT-4o mini
To find the volume of the solid, we first need to calculate the volume of both the cone and the hemisphere separately, and then add them together.
1. **Volume of the Hemisphere:**
The volume \( V_h \) of a hemisphere is given by:
\[
V_h = \frac{2}{3} \pi r^3
\]
where \( r \) is the radius of the hemisphere. In this case, the radius \( r = 1 \) cm. Substituting this value into the formula:
\[
V_h = \frac{2}{3} \pi (1)^3 = \frac{2}{3} \pi \text{ cm}^3
\]
2. **Volume of the Cone:**
The volume \( V_c \) of a cone is given by:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone. In this case, the radius \( r = 1 \) cm, and the height \( h = 1 \) cm (since the height of the cone is equal to its radius). Substituting these values into the formula:
\[
V_c = \frac{1}{3} \pi (1)^2 (1) = \frac{1}{3} \pi \text{ cm}^3
\]
3. **Total Volume of the Solid:**
The total volume \( V_{total} \) of the solid is the sum of the volume of the hemisphere and the volume of the cone:
\[
V_{total} = V_h + V_c
\]
Now we can substitute the volumes we calculated:
\[
V_{total} = \frac{2}{3} \pi + \frac{1}{3} \pi
\]
Combining these fractions:
\[
V_{total} = \left(\frac{2}{3} + \frac{1}{3}\right) \pi = \frac{3}{3} \pi = \pi \text{ cm}^3
\]
Thus, the total volume of the solid in terms of \( \pi \) is:
\[
\boxed{\pi} \text{ cm}^3
\]
1. **Volume of the Hemisphere:**
The volume \( V_h \) of a hemisphere is given by:
\[
V_h = \frac{2}{3} \pi r^3
\]
where \( r \) is the radius of the hemisphere. In this case, the radius \( r = 1 \) cm. Substituting this value into the formula:
\[
V_h = \frac{2}{3} \pi (1)^3 = \frac{2}{3} \pi \text{ cm}^3
\]
2. **Volume of the Cone:**
The volume \( V_c \) of a cone is given by:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone. In this case, the radius \( r = 1 \) cm, and the height \( h = 1 \) cm (since the height of the cone is equal to its radius). Substituting these values into the formula:
\[
V_c = \frac{1}{3} \pi (1)^2 (1) = \frac{1}{3} \pi \text{ cm}^3
\]
3. **Total Volume of the Solid:**
The total volume \( V_{total} \) of the solid is the sum of the volume of the hemisphere and the volume of the cone:
\[
V_{total} = V_h + V_c
\]
Now we can substitute the volumes we calculated:
\[
V_{total} = \frac{2}{3} \pi + \frac{1}{3} \pi
\]
Combining these fractions:
\[
V_{total} = \left(\frac{2}{3} + \frac{1}{3}\right) \pi = \frac{3}{3} \pi = \pi \text{ cm}^3
\]
Thus, the total volume of the solid in terms of \( \pi \) is:
\[
\boxed{\pi} \text{ cm}^3
\]