Examkin

Question

Difficulty: MediumThree-Dimensional Geometry: Volume and Surface Area

A solid right circular cylinder has a base radius of rr and a height of 4r4r. A solid sphere has a radius of RR. If the total surface area of the sphere is equal to the total surface area of the cylinder, what is the ratio of the volume of the sphere to the volume of the cylinder?

  1. 51012\frac{5\sqrt{10}}{12}Answer
  2. B
    102\frac{\sqrt{10}}{2}
  3. C
    223\frac{2\sqrt{2}}{3}
  4. D
    5106\frac{5\sqrt{10}}{6}
  5. E
    53\frac{5}{3}

Answer

The ratio of the volume of the sphere to the volume of the cylinder is 51012\frac{5\sqrt{10}}{12}.
The total surface area of the cylinder is the sum of its lateral area and two circular bases: 2πr(4r)+2πr2=10πr22\pi r(4r) + 2\pi r^2 = 10\pi r^2. Setting this equal to the sphere's surface area 4πR24\pi R^2 yields R/r=5/2=10/2R/r = \sqrt{5/2} = \sqrt{10}/2. The ratio of the sphere's volume 43πR3\frac{4}{3}\pi R^3 to the cylinder's volume πr2(4r)=4πr3\pi r^2 (4r) = 4\pi r^3 is 13(R/r)3=13(102)3=51012\frac{1}{3}(R/r)^3 = \frac{1}{3} \left(\frac{\sqrt{10}}{2}\right)^3 = \frac{5\sqrt{10}}{12}.

Step-by-Step Solution

1
Calculate the total surface area of the cylinder.
TSAcyl=2πr2+2πrh=2πr2+2πr(4r)=10πr2\text{TSA}_{\text{cyl}} = 2\pi r^2 + 2\pi r h = 2\pi r^2 + 2\pi r(4r) = 10\pi r^2
A cylinder's total surface area consists of two circular bases (2×πr22\times\pi r^2) and the lateral surface area (2πrh2\pi r h).
2
Equate the total surface area of the sphere to the total surface area of the cylinder to find the ratio of RR to rr.
4πR2=10πr2  ⟹  R2=52r2  ⟹  Rr=52=1024\pi R^2 = 10\pi r^2 \implies R^2 = \frac{5}{2}r^2 \implies \frac{R}{r} = \sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2}
The total surface area of a sphere of radius RR is 4πR24\pi R^2.
3
Express the volumes of both figures in terms of rr and RR.
Vsph=43πR3V_{\text{sph}} = \frac{4}{3}\pi R^3 and Vcyl=πr2h=πr2(4r)=4πr3V_{\text{cyl}} = \pi r^2 h = \pi r^2 (4r) = 4\pi r^3
The volume of a sphere is 43πR3\frac{4}{3}\pi R^3 and the volume of a cylinder is πr2h\pi r^2 h.
4
Compute the ratio of the volume of the sphere to the volume of the cylinder.
VsphVcyl=43πR34πr3=13(Rr)3=13(102)3=13⋅10108=51012\frac{V_{\text{sph}}}{V_{\text{cyl}}} = \frac{\frac{4}{3}\pi R^3}{4\pi r^3} = \frac{1}{3}\left(\frac{R}{r}\right)^3 = \frac{1}{3}\left(\frac{\sqrt{10}}{2}\right)^3 = \frac{1}{3} \cdot \frac{10\sqrt{10}}{8} = \frac{5\sqrt{10}}{12}
Substitute Rr=102\frac{R}{r} = \frac{\sqrt{10}}{2} into the ratio expression.

Key Concept

Relating 3D surface area formulas to volume formulas for cylinders and spheres
Estimated Time:2m 0s
Rate this question

Topics

Three-Dimensional Geometry: Volume and Surface Area

More questions from this topic