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Difficulty: MediumVolume and Surface Area of Solids

A right circular cylinder has a height of hh and a base radius of rr. A sphere has a radius of 2r2r. If the volume of the sphere is 88 times the volume of the cylinder, what is the value of the ratio hr\frac{h}{r}?

  1. A
    13\frac{1}{3}
  2. B
    34\frac{3}{4}
  3. 43\frac{4}{3}Answer
  4. D
    83\frac{8}{3}

Answer

43\frac{4}{3}
The volume of a sphere with radius RR is 43πR3\frac{4}{3}\pi R^3. Substituting R=2rR = 2r gives a volume of 43π(2r)3=323πr3\frac{4}{3}\pi (2r)^3 = \frac{32}{3}\pi r^3. The volume of a cylinder is πr2h\pi r^2 h. Since the volume of the sphere is 88 times the volume of the cylinder, we can set up the equation 323πr3=8πr2h\frac{32}{3}\pi r^3 = 8\pi r^2 h. Dividing both sides of the equation by 8πr28\pi r^2 yields 43r=h\frac{4}{3}r = h, which means the ratio hr\frac{h}{r} is 43\frac{4}{3}.

Step-by-Step Solution

1
Express the volume of both the cylinder and the sphere in terms of rr and hh.
The cylinder has volume Vcylinder=πr2hV_{\text{cylinder}} = \pi r^2 h. The sphere has radius 2r2r, so its volume is Vsphere=43π(2r)3=323πr3V_{\text{sphere}} = \frac{4}{3}\pi(2r)^3 = \frac{32}{3}\pi r^3.
This establishes formulas for both volumes using the given variables.
2
Set up the equation using the given relation that the volume of the sphere is 88 times the volume of the cylinder.
323πr3=8πr2h\frac{32}{3}\pi r^3 = 8\pi r^2 h
This represents the mathematical relationship specified in the problem statement.
3
Solve for the ratio hr\frac{h}{r} by dividing both sides of the equation by 8πr28\pi r^2.
323×8r=h  ⟹  43r=h  ⟹  hr=43\frac{32}{3 \times 8} r = h \implies \frac{4}{3}r = h \implies \frac{h}{r} = \frac{4}{3}
This isolates the ratio of the height to the radius to get the final result.

Key Concept

Using the formulas for the volume of a right circular cylinder and a sphere to solve for an unknown dimension ratio.
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Topics

Volume and Surface Area of Solids

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