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}](https://tex.z-dn.net/?f=%5Ctext%7BVolume%20of%20space%20%3D%20Volume%20of%20cylinder%20-%20Volume%20of%20cone%7D)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Ctext%7BVolume%20of%20cylinder%3D%20%7D%5Cpi%5Ctimes%20r%5E2%5Ctimes%20h%20%5C%5C%20%5Ctext%7BWhere%3A%7D%20%5C%5C%20r%3D%5Ctext%7B%20radius%20of%20the%20base%20of%20the%20cylinder%7D%20%5C%5C%20h%5Ctext%7B%20%3D%20height%20of%20the%20cylinder%7D%20%5Cend%7Bgathered%7D)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Ctext%7Bhypothenus%7D%5E2%3Dopposite%5E2%2Badjacent%5E2%20%5C%5C%2017%5E2%3Dh%5E2%2B8%5E2%20%5C%5C%20289%3Dh%5E2%2B64%20%5C%5C%20289-64%3Dh%5E2%20%5C%5C%20225%3Dh%5E2%20%5C%5C%20h%5E2%3D225%20%5C%5C%20h%3D%5Csqrt%5B%5D%7B225%7D%20%5C%5C%20h%3D15m%20%5Cend%7Bgathered%7D)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Ctext%7BVolume%20of%20cylinder%20%3D%20%7D%5Cpi%5Ctimes%20r%5E2%5Ctimes%20h%20%5C%5C%20%5Ctext%7BVolume%20of%20cylinder%20%3D%20%7D%5Cpi%5Ctimes8%5E2%5Ctimes15%20%5C%5C%20%5Ctext%7BVolume%20of%20cylinder%20%3D%20%7D%5Cpi%5Ctimes64%5E%7B%7D%5Ctimes15%20%5C%5C%20%5Ctext%7BVolume%20of%20cylinder%20%3D%20%7D%5Cpi%5Ctimes960%20%5C%5C%20%5Ctext%7BVolume%20of%20cylinder%20%3D%20960%7D%5Cpi%20m%5E3%20%5Cend%7Bgathered%7D)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Ctext%7BVolume%20of%20a%20cone%20%3D%20%7D%5Cfrac%7B1%7D%7B3%7D%5Ctimes%5Cpi%5Ctimes%20r%5E2%5Ctimes%20h%20%5C%5C%20%5Ctext%7BWhere%3A%7D%20%5C%5C%20r%5Ctext%7B%20%3D%20radius%20of%20the%20base%20of%20the%20cone%7D%20%5C%5C%20%5Ctext%7Bh%3D%20height%20of%20the%20cone%7D%20%5Cend%7Bgathered%7D)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Ctext%7BVolume%20of%20a%20cone%20%3D%20%7D%5Cfrac%7B1%7D%7B3%7D%5Ctimes%5Cpi%5Ctimes%20r%5E2%5Ctimes%20h%20%5C%5C%20%5Ctext%7BVolume%20of%20a%20cone%20%3D%20%7D%5Cfrac%7B1%7D%7B3%7D%5Ctimes%5Cpi%5Ctimes8%5E2%5Ctimes15%20%5C%5C%20%5Ctext%7BVolume%20of%20a%20cone%20%3D%20%7D%5Cfrac%7B1%7D%7B3%7D%5Ctimes%5Cpi%5Ctimes64%5E%7B%7D%5Ctimes15%20%5C%5C%20%5Ctext%7BVolume%20of%20a%20cone%20%3D%20%7D%5Cfrac%7B1%7D%7B3%7D%5Ctimes%5Cpi%5Ctimes960%3D%5Cfrac%7B960%7D%7B3%7D%5Ctimes%5Cpi%3D320%5Ctimes%5Cpi%20%5C%5C%20%5Ctext%7BVolume%20of%20a%20cone%20%3D%20320%7D%5Cpi%20m%5E3%20%5Cend%7Bgathered%7D)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Ctext%7BVolume%20of%20space%20%3D%20Volume%20of%20cylinder%20-%20Volume%20of%20cone%7D%20%5C%5C%20%5Ctext%7BVolume%20of%20space%20%3D%20960%7D%5Cpi%5Ctext%7B%20-%20%7D320%5Cpi%20%5C%5C%20%5Ctext%7BVolume%20of%20space%20%3D%20%7D640%5Cpi%20m%5E3%20%5Cend%7Bgathered%7D)
Correct answer: Option D