Question - Calculating Dimensions of Cones
mathothercalculating cone dimensionscone dimensions calculationcone geometry formulascone volume and area
Solution:
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 unitsUnknowns:- 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 units2nd row:Given:- Area of the Base = 144π square unitsTo find the radius, we know that Area of the Base = π * Radius^2, which means Radius = sqrt(Area of Base / π) = sqrt(144) = 12 unitsNow 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 unitsUnknowns:- 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 units4th row:Given:- Volume of Cone = 64π cubic units- Height = 12 unitsUnknowns:- 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 units5th 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 unitsPlease 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.
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