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Zorluk: OrtaVolume and Surface Area of Solids

A solid right circular cone has a base radius of rr and a height of hh. A right circular cylinder has a base radius that is twice the base radius of the cone, and a height that is three times the height of the cone. What is the ratio of the volume of the cone to the volume of the cylinder?

  1. 136\frac{1}{36}Cevap
  2. B
    118\frac{1}{18}
  3. C
    112\frac{1}{12}
  4. D
    16\frac{1}{6}

Cevap

The ratio of the volume of the cone to the volume of the cylinder is 136\frac{1}{36}
The volume of a right circular cone is given by Vcone=13πr2hV_{\text{cone}} = \frac{1}{3}\pi r^2 h. Since the cylinder has a radius that is twice that of the cone (2r2r) and a height that is three times that of the cone (3h3h), its volume is Vcylinder=π(2r)2(3h)=12πr2hV_{\text{cylinder}} = \pi (2r)^2(3h) = 12\pi r^2 h. Dividing the cone's volume by the cylinder's volume gives 13πr2h12πr2h=136\frac{\frac{1}{3}\pi r^2 h}{12\pi r^2 h} = \frac{1}{36}.

Adım Adım Çözüm

1
Write the formula for the volume of the cone.
Vcone=13πr2hV_{\text{cone}} = \frac{1}{3}\pi r^2 h
This establishes the volume of the first solid in terms of rr and hh.
2
Write the formula for the volume of the cylinder using the scaled dimensions.
Vcylinder=π(2r)2(3h)=12πr2hV_{\text{cylinder}} = \pi (2r)^2(3h) = 12\pi r^2 h
The cylinder has a radius of 2r2r and a height of 3h3h, and the volume formula is πR2H\pi R^2 H.
3
Calculate the ratio of the volume of the cone to the volume of the cylinder.
VconeVcylinder=13πr2h12πr2h=136\frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{\frac{1}{3}\pi r^2 h}{12\pi r^2 h} = \frac{1}{36}
This compares the two volumes directly by dividing the cone's volume by the cylinder's volume.

Anahtar Kavram

Volume of cylinders and cones and dimensional scaling relationships.

Alternatif Yöntem

Choose convenient sample values for the variables, such as r=1r = 1 and h=3h = 3. The volume of the cone is Vcone=13π(1)2(3)=πV_{\text{cone}} = \frac{1}{3}\pi (1)^2 (3) = \pi. The cylinder has radius 2(1)=22(1) = 2 and height 3(3)=93(3) = 9, so its volume is Vcylinder=π(2)2(9)=36πV_{\text{cylinder}} = \pi (2)^2 (9) = 36\pi. The ratio of the cone's volume to the cylinder's volume is π36π=136\frac{\pi}{36\pi} = \frac{1}{36}.
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Volume and Surface Area of Solids

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