Question

Find the volume of the space inside the cylinder that is not filled by the cone using the image below.

Answer 1

To find the volume of the space inside the cylinder that is not filled by the cone, we proceed as follows:

Step 1: Establish the relationship which will enable you obtain the volume, as below:

$\text{Volume of space = Volume of cylinder - Volume of cone}$

Step 2: Calculate the volume of the cylinder

The volume of a cylinder is given by the formula as below:

$\begin{gathered} \text{Volume of cylinder= }\pi\times r^2\times h \\ \text{Where:} \\ r=\text{ radius of the base of the cylinder} \\ h\text{ = height of the cylinder} \end{gathered}$

Now,

radius of the cylinder = (diameter)/2 = AC/2 = 16/2 = 8m

height of cylinder: ?

The height of the cylinder is gotten as follows:

We now apply the Pythagorean theorem to obtain the value of h, as follows:

$\begin{gathered} \text{hypothenus}^2=opposite^2+adjacent^2 \\ 17^2=h^2+8^2 \\ 289=h^2+64 \\ 289-64=h^2 \\ 225=h^2 \\ h^2=225 \\ h=\sqrt[]{225} \\ h=15m \end{gathered}$

Therefore, the height of the cylinder is 15m

Therefore:

$\begin{gathered} \text{Volume of cylinder = }\pi\times r^2\times h \\ \text{Volume of cylinder = }\pi\times8^2\times15 \\ \text{Volume of cylinder = }\pi\times64^{}\times15 \\ \text{Volume of cylinder = }\pi\times960 \\ \text{Volume of cylinder = 960}\pi m^3 \end{gathered}$

Step 3: Calculate the volume of the cone

The volume of a cone is given by the formula as below:

$\begin{gathered} \text{Volume of a cone = }\frac{1}{3}\times\pi\times r^2\times h \\ \text{Where:} \\ r\text{ = radius of the base of the cone} \\ \text{h= height of the cone} \end{gathered}$

Now,

radius of the cone = (diameter)/2 = AC/2 = 16/2 = 8m

height of cone: 15m

Therefore:

$\begin{gathered} \text{Volume of a cone = }\frac{1}{3}\times\pi\times r^2\times h \\ \text{Volume of a cone = }\frac{1}{3}\times\pi\times8^2\times15 \\ \text{Volume of a cone = }\frac{1}{3}\times\pi\times64^{}\times15 \\ \text{Volume of a cone = }\frac{1}{3}\times\pi\times960=\frac{960}{3}\times\pi=320\times\pi \\ \text{Volume of a cone = 320}\pi m^3 \end{gathered}$

Finally, the volume of the space inside the cylinder that is not filled by the cone is:

$\begin{gathered} \text{Volume of space = Volume of cylinder - Volume of cone} \\ \text{Volume of space = 960}\pi\text{ - }320\pi \\ \text{Volume of space = }640\pi m^3 \end{gathered}$

Correct answer: Option D