Sphere Properties: Surface Area and Volume Ratios
Let the radii of the two spheres be $r_1$ and $r_2$.
Surface Area Ratio Calculation
The formula for the curved surface area of a sphere is $A = 4 \pi r^2$.
The ratio of the curved surface areas of the two spheres is given as $1 : 4$. Therefore:
$ \frac{A_1}{A_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2 $Given that $\frac{A_1}{A_2} = \frac{1}{4}$, we have:
$ \left(\frac{r_1}{r_2}\right)^2 = \frac{1}{4} $Taking the square root of both sides gives the ratio of the radii:
$ \frac{r_1}{r_2} = \sqrt{\frac{1}{4}} = \frac{1}{2} $So, the ratio of the radii $r_1 : r_2$ is $1 : 2$.
Volume Ratio Calculation
The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$.
The ratio of the volumes of the two spheres is:
$ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \left(\frac{r_1}{r_2}\right)^3 $Substituting the ratio of the radii ($ \frac{r_1}{r_2} = \frac{1}{2} $) calculated previously:
$ \frac{V_1}{V_2} = \left(\frac{1}{2}\right)^3 = \frac{1^3}{2^3} = \frac{1}{8} $Thus, the ratio of the volumes of the two spheres is $1 : 8$.
Conclusion
The ratio of the volumes is $1 : 8$, which corresponds to Option C.
Similar Questions
- The radius of a sphere is 16 cm. Find its surface area.
- Five solid cubes, each of volume $39304 \text{ cm}^3$, are joined end to end to form a cuboid. What is the lateral surface area (in $\text{cm}^2$) of the cuboid?
- What is the surface area of a sphere of radius 8 cm? (use $\pi = 3.14$)
- The ratio of the volumes of two cubes is 64 : 1331. What is the ratio of their total surface areas?
- The diameter of a hemisphere is 7 cm. What will its total surface area be?
- The diameters of the bases of two cones are equal. If their slant heights are in the ratio of 3 : 4, then what will the ratio of their curved surface areas be?
- A cube of volume 324 cm$^3$ is molded into a cuboid whose length, width and height are in the ratio $3 : 2 : 2$. Find the length of the cuboid.
- An aquarium is in the form of a cuboid whose external measures are $80 \text{ cm} \times 30 \text{ cm} \times 40 \text{ cm}$. The base, side faces and back face are to be covered with a coloured paper. Find the area of the paper needed?
- A godown is in the form of a cuboid of measures $60 \text{ m} \times 40 \text{ m} \times 30 \text{ m}$. How many cuboidal boxes can be stored in it if the volume of one box is $0.8 \text{ m}^3$?
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Important Questions from 3-D Mensuration
Find the total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
