5. A sphere and a right circular cylinder have equal volumes. The radius of the sphere equals the radius of the base of the cylinder. Compare the surface area of the sphere with the area of the total surface of the cylinder.

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04-11-2022
Mathematics

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5. A sphere and a right circular cylinder have equal volumes. The radius of the sphere equals the radius of the base of the cylinder. Compare the surface area of the sphere with the area of the total surface of the cylinder.

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AaziyahE493044 AaziyahE493044 · Answer 1 · 2022-11-04T00:40:40+01:00

Given:

The volume of sphere is equal to the volume of cylinder.

The radius of sphere is equal to the radius of the cylinder.

The objective is to compare the surface area of sphere with the total surface area of cylinder.

Explanation:

The general formula of total surfacea area of cylnder is,

[tex]\begin{gathered} A_C=2\pi r^2+2\pi rh \\ A_C=2\pi r(r+h)\text{ . . . . . . .(1)} \end{gathered}[/tex]

SImilarly, the general formula of surface area of sphere is,

[tex]A_S=4\pi r^2\text{ . . . . . . (2)}[/tex]

Now, to compare the surface area of two given shapes, divide equation (2) by (1).

[tex]\begin{gathered} \frac{A_S}{A_C}=\frac{4\pi r^2_{}}{2\pi r(r+h)} \\ \frac{A_S}{A_C}=\frac{2r}{(r+h)}\text{ . . . . . (3)} \end{gathered}[/tex]

To find (h/r):

Since, the volume of sphere is equal to the volume of cylinder,

[tex]\begin{gathered} \frac{4}{3}\pi r^3=\pi r^2h \\ \frac{4}{3}=\frac{\pi r^2h}{\pi r^3} \\ \frac{4}{3}=\frac{h}{r} \\ r=\frac{3h}{4}\text{ . . . . .(4)} \end{gathered}[/tex]

Now, substitute the value of equation (4) in equation (3).

[tex]\begin{gathered} \frac{A_S}{A_C}=\frac{2(\frac{3h}{4})}{(\frac{3h}{4}+h)} \\ \frac{A_S}{A_C}=\frac{6h}{4(\frac{3h+4h}{4})} \\ \frac{A_S}{A_C}=\frac{6h}{7h} \\ \frac{A_S}{A_C}=\frac{6}{7} \end{gathered}[/tex]