Week 10 Assignment: Applet 1

Discovering the Formula for the Volume of a Sphere

Instuctions:

In this activity you will be tasked with discovering the formula for the volume of a Sphere of Radius \(r\).

1. In the applet below, adjust the slider for \(r\) and note that the observations are recorded in the spreadsheet to the right.
2. Once you have several observations ranging from 1 to 5, press the "Open Spreadsheet button, and copy the recorded values onto the new spreadsheet.
• HINT: if your spreadsheet includes \(r\)=1 more than once, make sure to rewrite the area as 3.14159 instead of \(\pi\), this will help when copying data elsewhere
3. From this new spreadsheet, use your experience working with lists of data to experiment with the data to try to find a pattern in the data.

Once you have an idea of the pattern in volume, continue onto the Desmos Applet Below.

4. Now, copy the data from the list above into the the Desmos app, and it should display a table of data, possibly labeled \(x_1, y_1, \text{and } z_1\).
5. Next, try entering the formula y_1 ~ a x_1^2 (displayed as \(y_1 \sim ax_1^2\)) and see what happens.
• Change the previous input slightly to get y_1 ~ a pi x_1^2 (displayed as \(y_1 \sim a\pi x_1^2\))
• Additionally you might try entering the more general y_1 ~ a x_1^2 + bx_1 +c (displayed as \(y_1 \sim a x_1^2 + bx_1 +c\) )
6. Finally, use your observations from working with the data as well as the previous step to construct a function which describes the volume (\(z_1\)) in terms of the radius (\(x_1\)).