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Sagot :
Answer:
Height of the cylinder = 13.8 cm (nearest tenth)
Volume of the cylindrical part = 2572.1 cm³ (nearest tenth)
Step-by-step explanation:
The total surface area of the solid figure is given as 1100 cm². This comprises of:
- The curved surface area of the cylinder.
- The curved surface area of the hemisphere.
- The area of the circular base of the cylinder (since it is exposed at the bottom).
The circular base of the hemisphere and the top of the cylinder are not included in the surface area calculation because they are internal surfaces and therefore do not contribute to the exterior surface area.
Therefore, the total surface area of the solid figure is the sum of the surface area of the cylinder and the hemisphere less the area of two circular bases:
[tex]\textsf{Total surface area}=(2\pi r h + 2 \pi ^2) + (3\pi r^2) - (2 \pi r^2) \\\\\textsf{Total surface area}=2\pi r h + 3\pi r^2[/tex]
where r is the radius of the cylinder and hemisphere, and h is the height of the cylinder.
The total height of the solid figure is given as 21.5 cm. Since the hemisphere sits on top of the cylinder, the height of the cylinder (h) plus the radius of the hemisphere (r) must equal the total height:
[tex]h+r=21.5[/tex]
Rearrange the equation to isolate h:
[tex]h=21.5 - r[/tex]
Now, substitute h = 21.5 - r into the surface area expression, set it equal to 1100, and rearrange in the form ar² + br + c = 0
[tex]2\pi r (21.5-r)+ 3\pi r^2=1100 \\\\\\43\pi r - 2\pi r^2+3\pi r^2 = 1100\\\\\\43\pi r+\pi r^2=1100\\\\\\\pp r^2 + 43\pi r - 1100 = 0[/tex]
Solve for r using the quadratic formula:
[tex]r=\dfrac{-43\pi \pm \sqrt{(43\pi)^2 - 4(1)(-1100)}}{2(1)} \\\\\\ r=\dfrac{-43\pi \pm \sqrt{1849 \pi^2 +4400}}{2} \\\\\\ r=7.7035129602...,\;\;r=-142.791997...[/tex]
Since length is positive only, the radius is:
[tex]r=7.7035129602...\; \sf cm[/tex]
To find the height of the cylinder, substitute the found value of r into h = 21.5 - r:
[tex]h = 21.5 - 7.7035129602...\\\\h = 13.7964870...\\\\h=13.8\; \sf cm\;(nearest\;tenth)[/tex]
Therefore, the height of the cylinder is 13.8 cm (rounded to the nearest tenth).
To find the volume of the cylindrical part, substitute h = 21.5 - r and the exact value of r into the volume of a cylinder formula:
[tex]\textsf{Volume of the cylinder}=\pi r^2 h \\\\\textsf{Volume of the cylinder}=\pi r^2 (21.5-r) \\\\\textsf{Volume of the cylinder}=\pi (7.7035129602...)^2 (21.5-7.7035129602...) \\\\\textsf{Volume of the cylinder}=2572.1484208952...\\\\\textsf{Volume of the cylinder}=2572.1\; \sf cm^3\;(nearest\;tenth)[/tex]
Therefore, the volume of the cylindrical part is 2572.1 cm³ (rounded to the nearest tenth).
[tex]\dotfill[/tex]
Additional Notes
It is generally not good practice to use rounded values mid-calculation because it can introduce rounding errors and reduce the accuracy of the final result. Therefore, we have used the exact values of r in the calculation of the volume of the cylindrical part. If we were to use the rounded value of r (r = 7.7 cm), the volume would be 2570.5 cm³.
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