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Question 1198556: A block of wood is in the form of a right circular cone. The altitude is 12 cm and the radius of the base is 5 cm. A cylindrical hole 5 cm in diameter is bored completely through the solid, the axis of the hole coinciding with the axis of the cone. The amount of wood left after the hole is bored may be expressed as V = Χπ/γ cm^3 where Χ is a positive integer and γ is a prime number. Find ∛χ + γ.
Answer by onyulee(41)   (Show Source):
You can put this solution on YOUR website! Certainly, let's solve this problem.
**1. Find the Volume of the Cone**
* Volume of cone = (1/3) * π * r² * h
* Where:
* r = radius of the base (5 cm)
* h = height of the cone (12 cm)
* Volume of cone = (1/3) * π * (5 cm)² * 12 cm
* Volume of cone = 100π cm³
**2. Find the Volume of the Cylindrical Hole**
* Volume of cylinder = π * r² * h
* Where:
* r = radius of the hole (5 cm / 2 = 2.5 cm)
* h = height of the hole (12 cm)
* Volume of cylinder = π * (2.5 cm)² * 12 cm
* Volume of cylinder = 75π cm³
**3. Find the Volume of the Remaining Wood**
* Volume of remaining wood = Volume of cone - Volume of cylinder
* Volume of remaining wood = 100π cm³ - 75π cm³
* Volume of remaining wood = 25π cm³
**4. Express the Volume in the Given Form**
* V = Χπ/γ cm³
* Comparing:
* V = 25π cm³
* V = Χπ/γ cm³
* We can see that:
* Χ = 25 * γ
* Since γ is a prime number, the smallest possible value for γ is 11 (the next prime number after 7 is 11).
* Therefore:
* Χ = 25 * 11 = 275
**5. Calculate ∛Χ + γ**
* ∛Χ + γ = ∛275 + 11
* ∛Χ + γ ≈ 6.47 + 11
* ∛Χ + γ ≈ 17.47
**Therefore, ∛Χ + γ is approximately 17.47.**
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