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Answer:
To find the volume of the solid with a circular base of radius 10 cm where every plane perpendicular to a fixed diameter is a square, we can use the formula for the volume of a pyramid:
V = (1/3) × A × h
Where:
- V is the volume of the pyramid
- A is the area of the base
- h is the height of the pyramid
Given:
- Radius of the circular base (r) = 10 cm
- Every plane perpendicular to a fixed diameter is a square
Step 1: Calculate the area of the circular base.
Area of a circle = π × r²
Area of the circular base = π × (10 cm)² = 314.16 cm²
Step 2: Find the height of the pyramid.
Since every plane perpendicular to a fixed diameter is a square, the height of the pyramid is equal to the side length of the square.
Let the side length of the square be s.
The diagonal of the square is equal to the diameter of the circle.
Diagonal of a square = s × √2
Diameter of the circle = 2 × r = 20 cm
s × √2 = 20 cm
s = 20 cm / √2 = 14.14 cm
Step 3: Calculate the volume of the pyramid using the formula.
V = (1/3) × A × h
V = (1/3) × 314.16 cm² × 14.14 cm
V = 1477.78 cm³
Therefore, the volume of the solid is approximately 1477.78 cm³.