A missile is made of a conical head mounted on a cylindrical base, both having…
2026
A missile is made of a conical head mounted on a cylindrical base, both having the same radius. The density of the conical head is twice that of the cylindrical base. The base has density 100 kg/m3, and the missile has mass 594000/7 kg. The cylindrical base is 10 m long, while the total missile length is 15 m. Find the radius of the conical head. Use π = 22/7.
Answer: B. 4.5 metres — ConceptFor a composite solid with common radius r, total mass is the sum of density × volume for each part. Thus, for a cylinder and a cone, M = ρcylπr2hcyl +…
- A.
4 metres
- B.
4.5 metres
- C.
5.5 metres
- D.
5 metres
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Correct answer: B
Concept
For a composite solid with common radius r, total mass is the sum of density × volume for each part.
Thus, for a cylinder and a cone, M = ρcylπr2hcyl + ρcone(1/3)πr2hcone. Different densities must be applied to their respective volumes.
Application
The cone height is 15 − 10 = 5 m.
Let the base density be ρcyl = 100 kg/m3. Then the head density is ρcone = 2 × 100 = 200 kg/m3.
M = 100(πr2 × 10) + 200[(1/3)πr2 × 5] = 1000πr2 + (1000/3)πr2 = (4000/3)πr2.
Substitute M = 594000/7 and π = 22/7: 594000/7 = (4000/3)(22/7)r2. Cancelling 1/7 and simplifying gives 594000 = (88000/3)r2, so r2 = (594000 × 3)/88000 = 20.25.
A radius is positive, so r = √20.25 = 4.5 m.
Cross-check
At r = 4.5 m, the cylinder mass is 100π(20.25)(10) = 20250π kg and the cone mass is 200(1/3)π(20.25)(5) = 6750π kg. Their sum is 27000π = 594000/7 kg when π = 22/7, which matches the stated mass.
Therefore, the radius of the conical head is 4.5 metres.