Problem: Find the new volume of a sphere when its radius is doubled, given the initial volume is $288\pi \text{ cm}^3$. Correct Answer: $2304\pi \text{ cm}^3$.
Volume Scaling Principle
The volume ($V$) of a sphere is related to its radius ($r$) by the formula $V = \frac{4}{3}\pi r^3$. This means the volume is proportional to the cube of the radius ($V \propto r^3$).
If the radius is scaled by a factor $k$, the volume is scaled by a factor of $k^3$. In this problem, the radius is doubled, so the scaling factor $k=2$. Therefore, the volume is multiplied by $k^3 = 2^3 = 8$.
Calculating New Volume
Given: Initial Volume ($V_{initial}$) = $288\pi \text{ cm}^3$. The radius is doubled, so the scaling factor $k=2$.
Calculation: New Volume ($V_{new}$) = Initial Volume $\times k^3$ $V_{new} = 288\pi \times (2)^3$ $V_{new} = 288\pi \times 8$ $V_{new} = 2304\pi \text{ cm}^3$
Conclusion
When the radius of the sphere is doubled, the new volume is $2304\pi \text{ cm}^3$. This matches Option B.
Similar Questions
- The base area of a prism is increased by 25%, while the height remains the same. By what percentage does the volume increase?
- A pyramid with an equilateral triangular base of side 10 cm and height 14 cm is placed on top of a cube with side length 10 cm. Find the total volume of the composite solid.
- In a right prism with a square base, volume = $1024 \text{ cm}^3$ and height = 16 cm. What is the side of the square base?
- If the numerical value of the volume of a sphere is equal to the numerical value of its surface area, find the radius of the sphere.
- A water tank is shaped like a composite prism: the bottom part is a triangular prism with base area 60 cm² and height 1.2 m, while the top part is a rectangular prism with base 40 cm × 30 cm and height 0.8 m. What is the total volume of the tank in liters? (1 m³ = 1000 L)
- A company designs a new chocolate box in the shape of a regular right pyramid with a square base. The base side of the box is 10 cm and its height is 12 cm. Due to packaging constraints, the box can only be filled up to 90% of its total volume. If the company wants to estimate the total cost of chocolate, knowing that $1 \text{ cm}^3$ of chocolate costs ₹0.50, what is the cost of chocolate to fill one such box?
- A right prism has a base in the shape of a trapezium with parallel sides 10 cm and 6 cm, and height 4 cm. If the prism height is 15 cm, what is the volume?
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Important Questions from 3-D Mensuration
- Find the surface area of a sphere whose diameter is equal to 98 cm.
- A conical vessel has base radius 31 cm and height 45 cm. Water is poured into the vessel until it is $\frac{2}{3}$ full. Find the volume (in cm³) of water in the vessel.
- A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?
- Find the volume (in cm³) of the largest right circular cone that can be cut out from a cube with an edge of 8 cm. Use $\pi = \frac{22}{7}$
- The volume of a solid cylinder is 5852 cm³ and its height is 38 cm. What is the total surface area of the solid cylinder? (Round your answer to the nearest integer) (Use $\pi = \frac{22}{7}$)
