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

Find the volume of the space inside the cylinder that is not filled by the cone using-example-1
User Wubin Ouyang
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1 Answer

19 votes
19 votes

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* r^2* 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* r^2* h \\ \text{Volume of cylinder = }\pi*8^2*15 \\ \text{Volume of cylinder = }\pi*64^{}*15 \\ \text{Volume of cylinder = }\pi*960 \\ \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 = }(1)/(3)*\pi* r^2* 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 = }(1)/(3)*\pi* r^2* h \\ \text{Volume of a cone = }(1)/(3)*\pi*8^2*15 \\ \text{Volume of a cone = }(1)/(3)*\pi*64^{}*15 \\ \text{Volume of a cone = }(1)/(3)*\pi*960=(960)/(3)*\pi=320*\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

Find the volume of the space inside the cylinder that is not filled by the cone using-example-1
User OverStack
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