Answer:
125π√3/3 cm³ ≈ 226.72 cm³
Step-by-step explanation:
The length of the circular edge of the half-circle is ...
(1/2)C = (1/2)(2πr) = πr = 10π . . . . cm
This is the circumference of the circular edge of the cone, so the radius of the cone is found from ...
C = 2πr
10π = 2πr . . . . fill in the numbers; next, solve for r
r = 5 . . . . cm
The slant height of the cone is the original radius, 10 cm, so the height of the cone from base to apex is found from the Pythagorean theorem.
(10 cm)² = h² + r²
h = √((10 cm)² -(5 cm)²) = 5√3 cm
And the cone's volume is ...
V = 1/3·πr²h = (1/3)π(5 cm)²(5√3 cm)
V = 125π√3/3 cm³ ≈ 226.72 cm³
The volume of the cone is approximately 226.7 cm³
The reasoning for obtaining the volume of the cone is as follows;
The known parameters are;
The shape from which the cone is made = A half circle
The radius of the half circle from which the cone is made, R = 10 cm
The required parameter;
The volume of the cone
Method;
We note that the circumference of circular base of the cone = Half the circumference of a circle
The slant height of the cone, l = The radius of the circle, R
The volume of a cone, V = (1/3)·π·r²·h
Where;
r = The radius of the base of the cone
h = The height of the cone = √(R² - r²)
Therefore, we find r as follows;
The circumference of the cone base = 2·π·r
Half the circumference of the circle from which the one is made = π·R
Therefore;
2·π·r = π·R
r = R/2
∴ r = 10/2 = 5
r = 5 cm
h = √(R² - r²)
∴ h = √(10² - 5²) = 5·√3
The height of the cone, h = 5·√3
∴ V = (1/3) × π × 5² × (5·√3) = (125/3)·π·√3 ≈ 226.7
The volume of the cone, V ≈ 226.7 cm³
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