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Discovering Formulae

October 31, 2017

We looked at the formulae for volume and surface area of spheres today.

We used oranges and a grapefruit as our spheres.  First step was to determine the radius, diameter and circumference of the sphere.

The next step involved peeling it, and making the peels into a rectangle.  We measured the dimensions of the rectangle, and then calculated the area of it, using length times width.

The next step was to relate the values we found.  We compared the square of the diameter with the area, since we need to compare things with similar units.

Our groups had a variety of answers, but most of them were close to Pi times diameter squared is equal to the area of the sphere.

We used our algebra skills to show that (pi)(d^2) is the same thing as 4(pi)(r^2).

To determine the volume of a sphere we used a tennis ball a juice concentrate container and a displacement tank.  We cut the juice concentrate container so it had a height equal to its diameter (which is conveniently the same as the tennis ball).

We used the displacement tank, and submerged the tennis ball, and held it until the water stopped pouring out.  We collected the water in the juice container.

It filled the cylinder 2/3 of the way.

We used algebra skills to examine the formula for the cylinder (replaced height by 2r, since the height is equal to the diameter, which is twice the radius).  The new formula for cylinder volume is Pi(r^2)(2r) which becomes 2(pi)(r^3)

Since the sphere displaced water equal to 2/3 of the cylinder’s volume, the sphere’s volume is (2/3)(2)(Pi)(r^3) which simplifies to (4/3)(Pi)(r^3).

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