[tex]\bold{\huge{\underline{ Solution }}}[/tex]
Given :-
- The length of the part is 16 centimeters
- The part of new engine is composed of 1 cone, 1 cylinder and 1 hemisphere
- The width of the engine is 6 cm
- The height of the cone is 5 cm
To Find :-
- We have to find the total volume of the part
Let's Begin :-
Let divide the part of engine into three parts as it is composed of 3 different figures.
We know that,
Volume of cone
[tex]\bold{=}{\bold{\dfrac{ 1}{3}}}{\bold{{\pi}r^{2}h}}[/tex]
Here, we have,
- The height of the cone is 5 cm
- The diameter of the cone is 6 cm
- Therefore,
- Radius of the cone = 3 cm
Subsitute the required values,
Volume of the first part that is cone
[tex]\sf{=}{\sf{\dfrac{ 1}{3}}}{\sf{ {\times}3.14{\times}(3)^{2}{\times}5}}[/tex]
[tex]\sf{=}{\sf{\dfrac{ 1}{3}}}{\sf{ {\times}3.14{\times}9{\times}5}}[/tex]
[tex]\sf{ = 3.14 {\times} 3 {\times} 5 }[/tex]
[tex]\sf{ = 3.14 {\times} 15 }[/tex]
[tex]\sf{ = 3.14 {\times} 15 }[/tex]
[tex]\bold{ = 47.1 cm^{3} }[/tex]
Thus, The volume of cone is 47.1 cm³ .
For second part
- Second part is composed of cylinder
We know that,
The volume of cylinder
[tex]\bold{ = {\pi}r^{2}h }[/tex]
Here,
- The diameter of the cylinder is 6 cm
- So, Radius = 3 cm
- The length of the cylinder = 16 - (Length of cone + Length of hemisphere)
- Length = 16 - 11 = 5 cm
Subsitute the required values in the above formula,
Volume of the second part that is cylinder
[tex]\sf{ = 3.14{\times} (3)^{2}{\times} 5}[/tex]
[tex]\sf{ = 3.14{\times} 9 {\times} 5}[/tex]
[tex]\sf{ = 3.14{\times} 45 }[/tex]
[tex]\bold{ = 141.3 cm^{3}}[/tex]
Thus, The volume of the cylinder is 141.3 cm³
For third part
- Third part is composed of hemisphere
We know that,
Volume of hemisphere
[tex]\bold{=}{\bold{\dfrac{ 2}{3}}}{\sf{ {\pi}r}}[/tex]
Here,
- The diameter of the hemisphere is 6 cm
- So, Radius = 3cm
Subsitute the required values,
Volume of third part that is hemisphere
[tex]\sf{=}{\sf{\dfrac{ 2}{3}}}{\sf{{\times} 3.14 {\times}3}}[/tex]
[tex]\sf{ = 2 {\times} 3.14 }[/tex]
[tex]\bold{ = 6.28 cm^{3} }[/tex]
Thus, The volume of the hemisphere is 6.28 cm³
Therefore,
The total volume of the part
= Volume of cone + Volume of cylinder + Volume of hemisphere
[tex]\sf{ = 47.1 + 141.3 + 6.28 }[/tex]
[tex]\sf{ = 188.4 + 6.28 }[/tex]
[tex]\sf{ = 194.68 cm^{3} }[/tex]
[tex]\bold{ = 194.7 cm^{3} }[/tex]
Hence, The total volume of the part is 194.7 cm³ .
