Drag and drop an answer to each box to correctly complete the explanation for deriving the formula for the volume of a sphere. For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with radius r and height 2r minus the volume of two cones each with radius r and height r. A cross section of the sphere is a circle and a cross section of the cylinder minus the cones, taken parallel to the base of the cylinder, is an annulus. The volume of the cylinder with radius r and height 2r is 2Ï€r^3. The volume of each cone with radius r and height r is (1/3)Ï€r^3. So the volume of the cylinder minus the two cones is (2Ï€r^3) – 2((1/3)Ï€r^3). Therefore, the volume of the cylinder is (2Ï€r^3) – (2/3)Ï€r^3, which simplifies to (4/3)Ï€r^3 by Cavalieri’s principle
The Correct Answer and Explanation is:
The volume of a sphere is derived by comparing it to a cylinder with two cones removed from it, using the concept of cross-sections and Cavalieri’s principle.
Step-by-Step Explanation:
- Cross Section of a Sphere:
- A cross-section of a sphere, when sliced through any plane parallel to its base, produces a circle. The radius of this circle varies depending on the height at which the sphere is sliced.
- Cross Section of a Cylinder:
- The cross-section of a cylinder with radius rrr and height 2r2r2r, when sliced parallel to its base, produces an annulus (a ring-shaped figure). The annulus represents the area between the outer boundary of the cylinder and the area inside the cross-section of the two cones.
- Volume of the Cylinder:
- The cylinder has a radius of rrr and height 2r2r2r. The formula for the volume of a cylinder is V=πr2hV = \pi r^2 hV=πr2h, so the volume of the cylinder is: Volume of cylinder=2πr3\text{Volume of cylinder} = 2\pi r^3Volume of cylinder=2πr3
- Volume of the Cones:
- Two cones are removed from the cylinder. Each cone has a radius rrr and height rrr. The formula for the volume of a cone is: Vcone=13πr2hV_{\text{cone}} = \frac{1}{3} \pi r^2 hVcone=31πr2h Substituting the height h=rh = rh=r, we get: Vcone=13πr3V_{\text{cone}} = \frac{1}{3} \pi r^3Vcone=31πr3 Since there are two cones, the total volume of the two cones is: Total volume of cones=2×13πr3=23πr3\text{Total volume of cones} = 2 \times \frac{1}{3} \pi r^3 = \frac{2}{3} \pi r^3Total volume of cones=2×31πr3=32πr3
- Subtracting the Volume of the Cones from the Cylinder:
- The volume of the sphere is the volume of the cylinder minus the volume of the two cones: Volume of sphere=2πr3−23πr3\text{Volume of sphere} = 2\pi r^3 – \frac{2}{3} \pi r^3Volume of sphere=2πr3−32πr3 Simplifying this expression: Volume of sphere=(2πr3−23πr3)\text{Volume of sphere} = \left( 2\pi r^3 – \frac{2}{3} \pi r^3 \right)Volume of sphere=(2πr3−32πr3) =63πr3−23πr3= \frac{6}{3} \pi r^3 – \frac{2}{3} \pi r^3=36πr3−32πr3 =43πr3= \frac{4}{3} \pi r^3=34πr3
- Final Formula:
- Therefore, the volume of the sphere is: V=43πr3V = \frac{4}{3} \pi r^3V=34πr3
This derivation uses Cavalieri’s principle, which states that two objects with the same cross-sectional area at every corresponding height must have the same volume. By comparing the sphere to a cylinder with cones removed, we can conclude that the volume of the sphere is 43πr3\frac{4}{3} \pi r^334πr3.
