Question

Reflect: Why was the volume of your fountain smaller than the volume of the ideal sphere? Discuss a more accurate method for approximating the volume of the spherical slab other than using just cylindrical slabs. Discuss in two to three sentences.

My fountain volume is: 454 cm^3 and my ideal sphere is 524 cm^3

Answer 1

The volume of a sphere is; 4/3•π•r³, therefore;

• The continuous changing radius of the sphere from the center, and the stepwise change in the radius of cylinder results in the difference in volume

• From the formula for a sphere, a more accurate method of approximating the volume is summing 4 cones, of the same radius as the sphere, and height equal to the radius

How can the volume of the sphere be approximated?

• The volume of the sphere obtained by approximation using cylinders, consist of several layers of cylinders that form steps which leaves spaces adjacent to the sphere on both sides

The volume of a sphere is presented as follows;

$v _s = \frac{4}{3} \times \pi \times {r}^{3}$

The volume of a cylinder is presented as follows;

$v _{cylinder} = \pi \times {r}^{2} \times h$

The volume of a cone is presented as follows;

$v _{cone} = \frac{1}{3} \times \pi \times {r}^{2} \times h$

Therefore;

• The volume of a sphere can be better approximated by summing-up the volumes of 4 cones of height equal to the base radius, which is the same as the radius of the sphere.

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