Answer :
Answer:
[tex] \sf V_{\textsf{cylinder}} :V_{\textsf{sphere}}= \dfrac{3}{2} [/tex]
[tex] \sf SA_{\textsf{cylinder}}:SA_{\textsf{sphere}} = \dfrac{3}{2} [/tex]
Step-by-step explanation:
To solve for the ratios of the volumes and surface areas of a cylinder and an inscribed sphere, we can use the following approach.
Let's denote:
[tex]\sf V_{\textsf{cylinder}} [/tex] as the volume of the cylinder,
- [tex]\sf V_{\textsf{sphere}} [/tex] as the volume of the inscribed sphere,
- [tex]\sf SA_{\textsf{cylinder}} [/tex] as the surface area of the cylinder,
- [tex]\sf SA_{\textsf{sphere}} [/tex] as the surface area of the inscribed sphere,
- [tex]\sf r [/tex] as the radius of the base of the cylinder and the radius of the inscribed sphere,
- [tex]\sf h [/tex] as the height of the cylinder.
Volume of the Cylinder [tex]\sf( V_{\textsf{cylinder}} )[/tex]:
The volume of a cylinder is given by:
[tex]\large\boxed{\boxed{ \sf V_{\textsf{cylinder}} = \pi r^2 h}} [/tex]
Volume of the Sphere [tex]\sf( V_{\textsf{sphere}} )[/tex]:
The volume of a sphere is given by:
[tex]\large\boxed{\boxed{ \sf V_{\textsf{sphere}} = \dfrac{4}{3} \pi r^3}} [/tex]
Surface Area of the Cylinder [tex]\sf( SA_{\textsf{cylinder}} )[/tex]:
The surface area of a cylinder is the sum of the areas of its two circular bases and its lateral surface area:
[tex]\large\boxed{\boxed{\sf SA_{\textsf{cylinder}} = 2 \pi r^2 + 2 \pi r h}} [/tex]
Surface Area of the Sphere [tex]\sf( SA_{\textsf{sphere}} )[/tex]:
The surface area of a sphere is given by:
[tex]\large\boxed{\boxed{ \sf SA_{\textsf{sphere}} = 4 \pi r^2}} [/tex]
Now let's find the ratios:
Note that: the height of the cone is twice the radius of the sphere.
Mathematically:
[tex]\sf h = 2 r [/tex]
Ratio of Volumes [tex]\sf\left( \dfrac{V_{\textsf{cylinder}}}{V_{\textsf{sphere}}} \right)[/tex]:
[tex]\begin{aligned}\sf \dfrac{V_{\textsf{cylinder}}}{V_{\textsf{sphere}}} & = \dfrac{\pi r^2 h}{\dfrac{4}{3} \pi r^3}\\\\ & = \dfrac{3h}{4r} \textsf{ (substitute h = 2r)} \\\\ & = \dfrac{3\cdot 2r}{4r} \\\\ & = \dfrac{6r}{4r} \\\\ & =\dfrac{3}{2} \end{aligned}[/tex]
Ratio of Surface Areas [tex]\sf\left( \dfrac{SA_{\textsf{cylinder}}}{SA_{\textsf{sphere}}} \right)[/tex]:
[tex]\begin{aligned}\sf \dfrac{SA_{\textsf{cylinder}}}{SA_{\textsf{sphere}}} & = \dfrac{2 \pi r^2 + 2 \pi r h}{4 \pi r^2} \\\\ & = \dfrac{2r + 2h}{4r} \\\\ & = \dfrac{r + h}{2r} \textsf{ (substitute h = 2r)} \\\\ & = \dfrac{r+2r}{2r}\\\\ & = \dfrac{3r}{2r}\\\\ & = \dfrac{3}{2}\end{aligned}[/tex]
Therefore:
Ratio of Volumes [tex]\sf( \dfrac{V_{\textsf{cylinder}}}{V_{\textsf{sphere}}} )[/tex]:
[tex] \sf \dfrac{V_{\textsf{cylinder}}}{V_{\textsf{sphere}}} = \dfrac{3}{2} [/tex]
Ratio of Surface Areas [tex]\sf( \dfrac{SA_{\textsf{cylinder}}}{SA_{\textsf{sphere}}} )[/tex]:
[tex] \sf \dfrac{SA_{\textsf{cylinder}}}{SA_{\textsf{sphere}}} = \dfrac{3}{2} [/tex]
Answer:
[tex]V_{\rm cylinder}:V_{\rm sphere}=\boxed{\dfrac{3}{2}}[/tex]
[tex]SA_{\rm cylinder}:SA_{\rm sphere}=\boxed{\dfrac{3}{2}}[/tex]
Step-by-step explanation:
When a sphere is inscribed in a cylinder, the radius of the sphere is equal to the radius of the circular base of the cylinder. Additionally, the height of the cylinder equals the diameter of the sphere, which is equivalent to the diameter of the cylinder’s circular base.
[tex]\dotfill[/tex]
Ratio of Volumes
The formulas for the volume of a cylinder and a sphere are:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Cylinder}}\\\\V=\pi r^2 h\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Sphere}}\\\\V=\dfrac{4}{3}\pi r^3\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\end{array}}[/tex]
As the height of the cylinder is equal to the diameter of its base, we can rewrite the formula for its volume in terms of r only by substituting h = 2r:
[tex]V_{\rm cylinder} = \pi r^2 \cdot 2r\\\\V_{\rm cylinder}=2\pi r^3[/tex]
Therefore, the ratio of the volume of the cylinder to the volume of the sphere is:
[tex]V_{\rm cylinder}:V_{\rm sphere}\\\\\\\phantom{ww}2\pi r^3 : \dfrac{4}{3}\pi r^3[/tex]
Divide both sides by πr³:
[tex]2 : \dfrac{4}{3}[/tex]
Express as a fraction:
[tex]\dfrac{2}{\frac{4}{3}}=\dfrac{3}{2}[/tex]
Therefore:
[tex]V_{\rm cylinder}:V_{\rm sphere}=\boxed{\dfrac{3}{2}}[/tex]
[tex]\dotfill[/tex]
Ratio of Surface Areas
The formulas for the surface area of a cylinder and a sphere are:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Surface Area of a Cylinder}}\\\\SA=2\pi r^2 +2\pi r h\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$SA$ is the surface area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
[tex]\boxed{\begin{array}{l}\underline{\textsf{Surface Area of a Sphere}}\\\\SA=4\pi r^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$SA$ is the surface area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\end{array}}[/tex]
As the height of the cylinder is equal to the diameter of its base, we can rewrite the formula for its surface area in terms of r only by substituting h = 2r:
[tex]SA_{\rm cylinder} = 2\pi r^2 + 2\pi r \cdot 2r\\\\SA_{\rm cylinder}=2\pi r^2+4\pi r^2\\\\SA_{\rm cylinder}=6\pi r^2[/tex]
Therefore, the ratio of the surface area of the cylinder to the surface area of the sphere is:
[tex]SA_{\rm cylinder}:SA_{\rm sphere}\\\\\\\phantom{www}6\pi r^2 : 4\pi r^2[/tex]
Divide both sides by πr²:
[tex]6:4[/tex]
Express as a fraction:
[tex]\dfrac{6}{4}=\dfrac{3}{2}[/tex]
Therefore:
[tex]SA_{\rm cylinder}:SA_{\rm sphere}=\boxed{\dfrac{3}{2}}[/tex]
[tex]\dotfill[/tex]
So, when a sphere is inscribed in a cylinder, the volume of the cylinder is one and a half times the volume of the sphere, and the surface area of the cylinder is also one and a half times the surface area of the sphere.
