Area and volume — sat-dog

Hard Math · Geometry and Trigonometry · Area and volume · MCQ

The figure shown is a right circular cylinder with a radius of $r$ and height of $h$ . A second right circular cylinder (not shown) has a volume that is $392$ times as large as the volume of the cylinder shown. Which of the following could represent the radius $R$ , in terms of $r$ , and the height $H$ , in terms of $h$ , of the second cylinder?

A
$upper R equals 8 r$ and $upper H equals 7 h$
B
$upper R equals 8 r$ and $upper H equals 49 h$
C
$upper R equals 7 r$ and $upper H equals 8 h$
D
$upper R equals 49 r$ and $upper H equals 8 h$

Correct answer: C

Explanation

Choice C is correct. The volume of a right circular cylinder is equal to $π a^{2} b$ , where $a$ is the radius of a base of the cylinder and $b$ is the height of the cylinder. It’s given that the cylinder shown has a radius of $r$ and a height of $h$ . It follows that the volume of the cylinder shown is equal to $π r^{2} h$ . It’s given that the second right circular cylinder has a radius of $R$ and a height of $H$ . It follows that the volume of the second cylinder is equal to $π R^{2} H$ . Choice C gives $upper R equals 7 r$ and $upper H equals 8 h$ . Substituting $7 r$ for $R$ and $8 h$ for $H$ in the expression that represents the volume of the second cylinder yields $pi left parenthesis 7 r right parenthesis squared left parenthesis 8 h right parenthesis$ , or $pi left parenthesis 49 r squared right parenthesis left parenthesis 8 h right parenthesis$ , which is equivalent to $pi left parenthesis 392 r squared h right parenthesis$ , or $392 left parenthesis pi r squared h right parenthesis$ . This expression is equal to $392$ times the volume of the cylinder shown, $π r^{2} h$ . Therefore, $upper R equals 7 r$ and $upper H equals 8 h$ could represent the radius $R$ , in terms of $r$ , and the height $H$ , in terms of $h$ , of the second cylinder.

Choice A is incorrect. Substituting $8 r$ for $R$ and $7 h$ for $H$ in the expression that represents the volume of the second cylinder yields $pi left parenthesis 8 r right parenthesis squared left parenthesis 7 h right parenthesis$ , or $pi left parenthesis 64 r squared right parenthesis left parenthesis 7 h right parenthesis$ , which is equivalent to $pi left parenthesis 448 r squared h right parenthesis$ , or $448 left parenthesis pi r squared h right parenthesis$ . This expression is equal to $448$ , not $392$ , times the volume of the cylinder shown.

Choice B is incorrect. Substituting $8 r$ for $R$ and $49 h$ for $H$ in the expression that represents the volume of the second cylinder yields $pi left parenthesis 8 r right parenthesis squared left parenthesis 49 h right parenthesis$ , or $pi left parenthesis 64 r squared right parenthesis left parenthesis 49 h right parenthesis$ , which is equivalent to $pi left parenthesis 3,136 r squared h right parenthesis$ , or $3,136 left parenthesis pi r squared h right parenthesis$ . This expression is equal to $3,136$ , not $392$ , times the volume of the cylinder shown.

Choice D is incorrect. Substituting $49 r$ for $R$ and $8 h$ for $H$ in the expression that represents the volume of the second cylinder yields $pi left parenthesis 49 r right parenthesis squared left parenthesis 8 h right parenthesis$ , or $pi left parenthesis 2,401 r squared right parenthesis left parenthesis 8 h right parenthesis$ , which is equivalent to $pi left parenthesis 19,208 r squared h right parenthesis$ , or $19,208 left parenthesis pi r squared h right parenthesis$ . This expression is equal to $19,208$ , not $392$ , times the volume of the cylinder shown.

Source: None

id cb_question_bank:224e737b-b611-442b-af26-ef476da15a5b · question a07ed090 · JSON