If the radius of a sphere is increased to twice its original size, what is the ratio of the new surface area to the original surface area, as well as the ratio of the new volume to the original volume?
4 : 1 and 8 : 1
Step 1 – Recall the formulas:
\(\text{Surface area} = 4\pi r^2,\quad \text{Volume} = \tfrac{4}{3}\pi r^3\)
Step 2 – Apply the scaling. If r → 2r:
\(\dfrac{SA_{new}}{SA_{old}} = \dfrac{4\pi (2r)^2}{4\pi r^2} = \dfrac{4r^2}{r^2} = 4\)
\(\dfrac{V_{new}}{V_{old}} = \dfrac{(2r)^3}{r^3} = 8\)
Step 3 – Read off the ratios:
\(SA \text{ ratio} = 4 : 1,\quad V \text{ ratio} = 8 : 1\)
Hence the answer is option 2.
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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 = \(22\over7\)]
Which of the following is a geometrical figure with a three-dimensional geometry that has eight vertices and six rectangular faces?
- 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 $\pi = \frac{22}{7}$
