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Scaling in Three Dimensions: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

Two solids are known to be similar. The original total surface area is 150 square cm and volume is 125 cubic cm. The image volume is 1000 cubic cm. What is the image surface-area-to-volume ratio?

Answer choices for question 1 A $1.2\text{ cm}^{-1}$ B $2.4\text{ cm}^{-1}$ C $0.15\text{ cm}^{-1}$ D $0.6\text{ cm}^{-1}$
2

A similar image has surface-area-to-volume ratio $4/5$ of the original ratio, with both expressed in the same unit. What is its image-to-original volume factor?

Answer choices for question 2 A $64/125$ B $125/64$ C $25/16$ D $5/4$
3

A similar image has $69\%$ more total surface area than its original. What percentage of the original volume is the image volume?

Answer choices for question 3 A $169\%$ B $130\%$ C $219.7\%$ D $119.7\%$
4

A cube has edge 4 cm. Candidate A is a cube of edge 12 cm. Candidate B is a rectangular box 4 by 4 by 108 cm. Both candidates have the same volume. Which statement is correct?

Answer choices for question 4 A A is similar to the original; B is not. B Both are similar because their volumes are equal. C Only B is similar to the original. D Neither is similar because both have larger volume.
5

A closed right cylinder has radius 2 cm and height 8 cm. Its volume must become 4 times as large. Plan A resizes the entire cylinder uniformly. Plan B keeps the radius fixed and changes only height. Which pair gives the two new heights, in Plan A then Plan B order?

Answer choices for question 5 A $16\text{ cm}$ and $32\text{ cm}$ B $32\text{ cm}$ and $8\sqrt[3]{4}\text{ cm}$ C $8\sqrt[3]{4}\text{ cm}$ and $32\text{ cm}$ D $8\sqrt[3]{4}\text{ cm}$ and $16\text{ cm}$
6

Solids A and B are similar. Their surface-area ratio A:B is $4:9$ . Their combined volume is 665 cubic cm. What is B's volume?

Answer choices for question 6 A $152\text{ cm}^3$ B $460.38\text{ cm}^3$ C $399\text{ cm}^3$ D $513\text{ cm}^3$
7

A similarity takes an original solid to an image with $125/216$ of its volume. The image surface area is 250 square cm. What is the original surface area?

Answer choices for question 7 A $300\text{ cm}^2$ B $360\text{ cm}^2$ C $432\text{ cm}^2$ D $173.61\text{ cm}^2$
8

An original closed right cylinder and its image have corresponding radii 5 and 10 cm and heights 6 and 12 cm, respectively, in original:image order. Their total surface-area ratio is 1:4. Which image-to-original volume factor is consistent with all these measurements?

Answer choices for question 8 A $8$ B $4$ C $2$ D $16$
9

A closed rectangular box is 3 by 5 by 6 cm. Its height alone is multiplied by a positive factor $t$ , with the 3 by 5 cm base fixed. Which expression gives its new total surface area in square cm?

Answer choices for question 9 A $126t^2$ B $30+96t$ C $126t$ D $30t+96$
10

A positive uniform resize doubles a solid's surface-area-to-volume ratio. What happens to its total surface area?

Answer choices for question 10 A It is multiplied by $1/4$ . B It is multiplied by $2$ . C It is multiplied by $4$ . D It is multiplied by $1/8$ .
11

A solid is claimed to be a similar image of another with length factor $3/2$ . Its measured total surface area has factor $9/4$ , but its measured volume has factor $3$ . What can be concluded if all measurements are exact and corresponding boundaries match?

Answer choices for question 11 A The claim is supported because the surface factor matches. B The claim is supported because every measurement increased. C The measurements contradict the claimed similarity. D Changing the direction of the comparison makes all three factors consistent.
12

A solid has surface area 0.54 square m and volume 0.027 cubic m. What is its surface-area-to-volume ratio expressed per centimeter?

Answer choices for question 12 A $20\text{ cm}^{-1}$ B $2000\text{ cm}^{-1}$ C $0.002\text{ cm}^{-1}$ D $0.2\text{ cm}^{-1}$

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