A right circular cone has a volume of 24\text{ cubic centimeters}. A second right circular… — SAT

A right circular cone has a volume of $24\text{ cubic centimeters}$ . A second right circular cone has a base radius that is $3$ times the base radius of the first cone, and a height that is half the height of the first cone. What is the volume, in cubic centimeters, of the second cone?

A
36
108Cevap
C
216
D
324

Cevap

108

The volume of a cone is proportional to the square of its radius and directly proportional to its height. Tripling the radius increases the volume by a factor of

3^2 = 9

. Halving the height reduces the volume by a factor of

\frac{1}{2}

. Combining these changes scales the original volume by

9 \times \frac{1}{2} = 4.5

. Multiplying the original volume of

24

4.5

gives the correct volume of

108

Adım Adım Çözüm

Write down the formula for the volume of a right circular cone.

V = \frac{1}{3}\pi r^2 h

This establishes the relationship between volume, radius, and height.

Express the volume of the second cone (

V_2

) in terms of the dimensions of the first cone (

r

and

h

V_2 = \frac{1}{3}\pi (3r)^2 \left(\frac{1}{2}h\right)

The radius of the second cone is

3r

and the height is

\frac{1}{2}h

Simplify the expression for the second cone's volume to find the scale factor relative to the first cone.

V_2 = \frac{1}{3}\pi (9r^2) \left(\frac{1}{2}h\right) = \frac{9}{2} \left(\frac{1}{3}\pi r^2 h\right) = 4.5 V_1

Squaring the radius multiplier of 3 gives a factor of 9, and the height multiplier of 0.5 gives a factor of 0.5, resulting in a combined scaling factor of 4.5.

Calculate the final volume using the volume of the first cone (

V_1 = 24

V_2 = 4.5 \times 24 = 108

Multiplying the original volume of

24\text{ cm}^3

by the overall scale factor of

4.5

yields the new volume.

Anahtar Kavram

Dimensional scaling of three-dimensional solids

Soru

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