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Difficulty: HardThree-Dimensional Geometry: Volume and Surface Area

A hollow metallic spherical shell has an inner radius of 33 centimeters and an outer radius of rr centimeters, where r>3r > 3. If the spherical shell is melted down and completely recast into a solid right circular cylinder with base radius rr centimeters and height 77 centimeters, what is the value of rr?

  1. A
    4.54.5
  2. B
    5.255.25
  3. 66Answer
  4. D
    7.57.5
  5. E
    99

Answer

6
The volume of metal in the hollow spherical shell is 43π(r3−33)=43π(r3−27)\frac{4}{3}\pi(r^3 - 3^3) = \frac{4}{3}\pi(r^3 - 27). The volume of the recast cylinder is πr2h=7πr2\pi r^2 h = 7\pi r^2. Equating these volumes gives 43(r3−27)=7r2\frac{4}{3}(r^3 - 27) = 7r^2, which simplifies to 4r3−21r2−108=04r^3 - 21r^2 - 108 = 0. Factoring this cubic equation gives (r−6)(4r2+3r+18)=0(r - 6)(4r^2 + 3r + 18) = 0. Since the quadratic term has no real roots, the only real solution is r=6r = 6.

Step-by-Step Solution

1
Set up the formula for the volume of metal in the hollow spherical shell.
Vshell=43π(r3−33)=43π(r3−27)V_{\text{shell}} = \frac{4}{3}\pi \left(r^3 - 3^3\right) = \frac{4}{3}\pi \left(r^3 - 27\right)
The metal occupies only the region between the inner sphere of radius 3 cm and outer sphere of radius r cm.
2
Set up the formula for the volume of the recast solid right circular cylinder.
Vcylinder=πr2h=7πr2V_{\text{cylinder}} = \pi r^2 h = 7\pi r^2
The cylinder has base radius r cm and height 7 cm.
3
Equate the two volumes since no metal is lost during melting and recasting.
43π(r3−27)=7πr2\frac{4}{3}\pi \left(r^3 - 27\right) = 7\pi r^2
Conservation of volume during recasting.
4
Simplify the equation and solve for r.
4(r3−27)=21r2  ⟹  4r3−21r2−108=04(r^3 - 27) = 21r^2 \implies 4r^3 - 21r^2 - 108 = 0
Divide both sides by π\pi and multiply by 3 to clear the fraction.
5
Factor the cubic polynomial 4r3−21r2−108=04r^3 - 21r^2 - 108 = 0.
(r−6)(4r2+3r+18)=0  ⟹  r=6(r - 6)(4r^2 + 3r + 18) = 0 \implies r = 6
Testing r=6r = 6 gives 4(216)−21(36)−108=864−756−108=04(216) - 21(36) - 108 = 864 - 756 - 108 = 0. The quadratic factor 4r2+3r+184r^2 + 3r + 18 has a negative discriminant and produces no real roots.

Key Concept

Volume formulas for hollow spheres and right circular cylinders
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Topics

Three-Dimensional Geometry: Volume and Surface Area

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