The question above is not well arranged. Please find the well arranged question below for proper understanding.
Complete Question:
A cone fits inside a square pyramid as shown. For every cross section, the ratio of the area of the circle to the area of the square is StartFraction pi r squared Over 4 r squared EndFraction or StartFraction pi Over 4 EndFraction.
A cone is inside of a pyramid with a square base. The cone has a height of h and a radius of r. The pyramid has a base length of 2 r.
Since the area of the circle is StartFraction pi Over 4 EndFraction the area of the square, the volume of the cone equals
A. StartFraction pi Over 4 EndFraction the volume of the pyramid or StartFraction pi Over 4 EndFractionStartFraction pi Over 4 EndFraction (StartFraction (2 r) (h) Over 3 EndFraction) or One-sixthπrh.
B. StartFraction pi Over 4 EndFraction the volume of the pyramid or StartFraction pi Over 4 EndFractionStartFraction pi Over 4 EndFraction (StartFraction (2 r) squared (h) Over 3 EndFraction) or One-thirdπr²h.
C. StartFraction pi Over 2 EndFraction the volume of the pyramid or StartFraction pi Over 2 EndFraction or Two-thirdsπr²h.
D. StartFraction pi Over 2 EndFraction the volume of the pyramid or StartFraction pi Over 4 EndFraction or One-thirdπr²h.
Answer:
B. StartFraction pi Over 4 EndFraction the volume of the pyramid or StartFraction pi Over 4 EndFraction (StartFraction (2 r) squared (h) Over 3 EndFraction) or One-thirdπr²h = 1/3πr²h
Step-by-step explanation:
We have two geometric shapes in the question.
a) A cone and b) a square pyramid
The cone has a height of h and a radius of r. The pyramid has a base length of 2 r.
The volume of a cone =1/3πr²h
Where πr² = Area of the circle at the base of the cone
Hence, Volume of a cone = 1/3 × Area of the circular base of a cone × Height
The volume of a square pyramid = 1/3a²h
Where a² = Area of the square base of the pyramid
Hence, Volume of a square pyramid = 1/3 × Area of the square base of a pyramid × height(h)
Base area of a cone / Base area of a square pyramid = π/4
Base area of a circle = Base area of a pyramid × π/4
Volume of a cone = 1/3πr²h
Volume of a cone = 1/3 × Base area of a square pyramid × π/4 × h
Note that:
Volume of a square pyramid = 1/3a²h
= 1/3 × Base area of a square pyramid × height
Hence,
Volume of a cone = Volume of a square pyramid × π/4
= StartFraction pi Over 4 EndFraction the volume of the pyramid
Or
Where a = base length = 2r
Volume of the square pyramid = 1/3 × 2r² × h = 1/3 × 4r²h
Volume of a cone = Volume of a square pyramid × π/4
Substituting = 1/3 × 4 × r²× h × π/4
Volume of a cone = 1/3 πr²h
Or
Volume of a cone = Volume of a square pyramid × π/4
Volume of a square pyramid when base length is 2r = 1/3 × (2r)² × h = (2r²)h/3
Substituting (2r²)h/3 for volume of a square pyramid in volume of a cone , we have:
Volume of a cone = π/4 × 2r²h/3
=
StartFraction pi Over 4 EndFractionStartFraction pi Over 4 EndFraction (StartFraction (2 r) squared
Therefore, Option B is correct
