Question

3 the volume of the cylinder is 27π cubic units. write the volume of the cone and hemisphere. express the volumes in terms of π. object volume (cu. units) cone cylinder 27π hemisphere

Explanation:

To solve this, we assume the cylinder, cone, and hemisphere have the same radius \( r \) and the height of the cylinder \( h \) is equal to the radius (since often in such problems, the height of the cylinder is equal to the radius for the cone and hemisphere to have the same dimensions, though we can also infer from the volume of the cylinder). Let's first recall the formulas:

Step 1: Find the radius from the cylinder's volume

The volume of a cylinder is given by \( V_{\text{cylinder}}=\pi r^{2}h \). We know \( V_{\text{cylinder}} = 27\pi \). Let's assume the height \( h\) of the cylinder is equal to the radius \( r\) (a common case when dealing with cone and hemisphere with the same base radius and related height). So \( h = r \).

Substitute into the cylinder's volume formula:
\[
\pi r^{2}\times r=27\pi
\]
\[
\pi r^{3}=27\pi
\]
Divide both sides by \( \pi \):
\[
r^{3}=27
\]
Take the cube - root of both sides:
\[
r = 3
\]

Step 2: Volume of the cone

The volume of a cone is given by \( V_{\text{cone}}=\frac{1}{3}\pi r^{2}h \). Since for a cone with the same base radius as the cylinder and height equal to the radius of the cylinder (or in the case where the height of the cone is equal to the radius, because we found \( r = 3\) and if we assume the height of the cone \( h=r = 3\)):

Substitute \( r = 3\) and \( h = 3\) into the cone's volume formula:
\[
V_{\text{cone}}=\frac{1}{3}\pi\times(3)^{2}\times3
\]
\[
V_{\text{cone}}=\frac{1}{3}\pi\times9\times3
\]
\[
V_{\text{cone}} = 9\pi
\]

Step 3: Volume of the hemisphere

The volume of a sphere is \( V_{\text{sphere}}=\frac{4}{3}\pi r^{3}\), so the volume of a hemisphere is \( V_{\text{hemisphere}}=\frac{2}{3}\pi r^{3}\).

Substitute \( r = 3\) into the hemisphere's volume formula:
\[
V_{\text{hemisphere}}=\frac{2}{3}\pi\times(3)^{3}
\]
\[
V_{\text{hemisphere}}=\frac{2}{3}\pi\times27
\]
\[
V_{\text{hemisphere}} = 18\pi
\]

Answer:

The volume of the cone is \( 9\pi \) cubic units and the volume of the hemisphere is \( 18\pi \) cubic units.