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Subject Test Math 2

Question 17 of 28

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SAT Subject Test

The figure shows a right circular cone with the horizontal base at the bottom and the vertex on top. A circular plane is drawn between the vertex and the base and is parallel to the base.

The right circular cone above is sliced horizontally forming two pieces, each of which has the same height. What is the ratio of the volume of the smaller piece to the volume of the larger piece?

Select an Answer

1 half

Correct Answer: 
No

1 third

Correct Answer: 
No

1 over 6

Correct Answer: 
No

1 over 7

Correct Answer: 
Yes

1 over 8

Correct Answer: 
No

It is helpful to label the figure.

The figure shows a right circular cone with the horizontal base at the bottom and the vertex on top. The radius of the base is labeled 2 r, and the height of the cone is labeled 2 h. A circular plane between the vertex and the base of the cone is drawn parallel to the base of the cone and divides the cone into two pieces. The top piece is a smaller cone with base of radius labeled r and the height labeled h.

The top piece is a cone whose height h is one-half the height of the original cone 2 h. Using the properties of similar right triangles, you should realize the radii of these two cones must be in the same ratio. So if the top cone has radius r, the original cone has radius 2 r.

The volume of the top piece is equal to fraction numerator 1 over denominator 3 end fraction pi r to the power of 2 end exponent h. The volume of the bottom piece is equal to the volume of the original cone minus the volume of the top piece.

fraction numerator 1 over denominator 3 end fraction pi left parenthesis 2 r right parenthesis to the power of 2 end exponent left parenthesis 2 h right parenthesis minus fraction numerator 1 over denominator 3 end fraction pi r to the power of 2 end exponent h

equals fraction numerator 1 over denominator 3 end fraction pi left parenthesis 8 r to the power of 2 end exponent h minus r to the power of 2 end exponent h right parenthesis

equals fraction numerator 7 over denominator 3 end fraction pi r to the power of 2 end exponent h

The ratio of the volume of the smaller piece to the volume of the larger piece is fraction numerator fraction numerator 1 over denominator 3 end fraction pi r to the power of 2 end exponent h over denominator fraction numerator 7 over denominator 3 end fraction pi r to the power of 2 end exponent h end fraction equals fraction numerator 1 over denominator 7 end fraction.

Question Difficulty: 
hard