Calculating Dimensions of Cones
This image shows a table to be filled in with information about a cone's dimensions. Let's fill it out row by row:
1st row:
Given:
- Radius = 4 units
Unknowns:
- Diameter = 2 * Radius = 2 * 4 units = 8 units
- Area of the Base = π * Radius^2 = π * 4^2 square units = π * 16 square units ≈ 50.27 square units (Using π ≈ 3.14159)
- Height = 3 units
- Volume of Cone = (1/3) * Area of Base * Height = (1/3) * π * 16 * 3 cubic units ≈ 50.27 cubic units
2nd row:
Given:
- Area of the Base = 144π square units
To find the radius, we know that Area of the Base = π * Radius^2, which means Radius = sqrt(Area of Base / π) = sqrt(144) = 12 units
Now use this to find the unknowns:
- Diameter = 2 * Radius = 2 * 12 units = 24 units
- Height = 6 units
- Volume of Cone = (1/3) * Area of Base * Height = (1/3) * 144π * 6 = 288π cubic units ≈ 904.78 cubic units (Using π ≈ 3.14159)
3rd row:
Given:
- Volume of Cone = 200π cubic units
- Diameter = 20 units
Unknowns:
- Radius = Diameter / 2 = 20 units / 2 = 10 units
- Area of the Base = π * Radius^2 = π * 10^2 square units = 100π square units
- To find the Height, use the formula Volume of Cone = (1/3) * Area of Base * Height. Rearrange this formula to solve for Height as Volume of Cone / (Area of Base / 3):
Height = 200π cubic units / (100π square units / 3) = 6 units
4th row:
Given:
- Volume of Cone = 64π cubic units
- Height = 12 units
Unknowns:
- To find the Area of the Base, we rearrange the volume formula to solve for Area of Base as (Volume of Cone / Height) * 3:
Area of the Base = (64π cubic units / 12 units) * 3 = 16π square units
- Radius = sqrt(Area of Base / π) = sqrt(16) = 4 units
- Diameter = 2 * Radius = 2 * 4 units = 8 units
5th row:
Given:
- Height = 3 units
- Volume of Cone = 3.14 cubic units (assuming π ≈ 3.14 for calculation simplicity)
Unknowns:
- Use the volume formula to solve for the Area of the Base:
Area of the Base = (Volume of Cone / Height) * 3 = (3.14 cubic units / 3 units) * 3 ≈ 3.14 square units
- Radius = sqrt(Area of Base / π) ≈ sqrt(3.14 / 3.14) = 1 unit
- Diameter = 2 * Radius = 2 * 1 unit = 2 units
Please note that when filling out the table, I only provided approximate values for areas and volumes involving π, as I assumed π ≈ 3.14159 for calculation purposes.
