Geometry Table Completions
To complete the table, we will use the following relationships and formulas:
1. The diameter of a circle is twice the radius: \(d = 2r\).
2. The area of the base (which is a circle) is given by \(A = \pi r^2\).
3. The volume of a cone is given by \(V = \frac{1}{3} \pi r^2 h\), where \(h\) is the height of the cone.
Let's fill in the missing values for each row of the table:
For the second row:
- Given the diameter as 3 units, the radius \(r\) is half of that, so \(r = \frac{3}{2}\) units.
- The area of the base \(A\) is \(\pi r^2 = \pi (\frac{3}{2})^2 = \frac{9}{4} \pi\) square units.
- The given volume \(V\) is \(27\pi\) cubic units, we can use it to find the height \(h\):
\(V = \frac{1}{3} \pi r^2 h\)
\(27\pi = \frac{1}{3} \pi \left(\frac{3}{2}\right)^2 h\)
\(27\pi = \frac{1}{3} \pi \frac{9}{4} h\)
\(27\pi = \pi \frac{9}{4} \frac{1}{3} h\)
\(27 = \frac{9}{12} h\)
\(h = 27 \times \frac{12}{9}\)
\(h = 36\)
So the height is 36 units.
For the third row:
- Given the radius \(r = 10\) units, the diameter \(d\) is twice that, so \(d = 20\) units.
- The area of the base \(A\) is \(\pi r^2 = \pi (10)^2 = 100\pi\) square units.
- The height \(h\) is 12 units, as given, so no calculation is needed for that.
For the fourth row:
- Given the volume \(V = 3.14\) cubic units and height \(h = 3\) units, we can find the radius \(r\):
\(V = \frac{1}{3} \pi r^2 h\)
\(3.14 = \frac{1}{3} \pi r^2 \cdot 3\)
\(3.14 = \pi r^2\)
\(r^2 = \frac{3.14}{\pi}\)
Because \(\pi\) is approximately 3.14, \(r^2\) is approximately 1, hence
\(r \approx 1\) unit.
- The approximate diameter \(d\) is twice the radius, so \(d \approx 2\) units.
- The area of the base \(A\) is \(\pi r^2 = \pi (1)^2 = \pi\) square units, which we can approximate as 3.14 square units since the volume was given to this level of precision.
The completed table should look like this:
- Diameter: 2, 3, 20, 2
- Radius: 1, 1.5, 10, 1
- Area of the base: \(\pi\), \(2.25\pi\), \(100\pi\), \(\pi\)
- Height: 3, 36, 12, 3
- Volume of Cone: \(3.14\), \(27\pi\), \(400\pi\), \(3.14\)
