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02 Jan 2005, 23:28
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A sculptor carved a stone into a perfect cube with faces measuring 100 square inches. He then carved this stone into a smaller cube with exactly one-third the volume of the original cube. The edge of the smaller cube measures most nearly
A. 7 inches.
B. 6 inches.
C. 5 inches.
D. 4 inches.
E. 3 inches.
A. 7 inches.
B. 6 inches.
C. 5 inches.
D. 4 inches.
E. 3 inches.
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03 Jan 2005, 00:56
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The original cube has faces with area 100 square inches, so those faces must have sides of length 10. That would give the cube a total volume of 10 ^ 3 = 1,000 cubic inches.
Since the new cube is one third the volume of the original, it has volume 1,000 / 3 = 333.33 cubic inches.
Now we have to figure out what side length would give a cube a volume of 333.33 cubic inches. Lets try some numbers from the answer choices..
7^3
= 7 * 7 * 7
= 49 * 7 =343.
Too big, but close.
6^3
= 6 * 6 * 6
= 36 * 6
= 198
Way too small.
All the other choices are even smaller, so the closest answer is 7 and the answer is A.
NOTE: The thing to understand about problems like this is that in proportional three dimensional shapes, if the volume changes by some amount, the linear measures will always change by the cubic root of that amount. In this case, the volume of the sphere changed to be 1/3 of its original, so the linear change is the cubic root of 1/3. Multiplying 10 by the cubic root of 1/3 gives 6.9 which is almost 7. Our answer checks.
Since the new cube is one third the volume of the original, it has volume 1,000 / 3 = 333.33 cubic inches.
Now we have to figure out what side length would give a cube a volume of 333.33 cubic inches. Lets try some numbers from the answer choices..
7^3
= 7 * 7 * 7
= 49 * 7 =343.
Too big, but close.
6^3
= 6 * 6 * 6
= 36 * 6
= 198
Way too small.
All the other choices are even smaller, so the closest answer is 7 and the answer is A.
NOTE: The thing to understand about problems like this is that in proportional three dimensional shapes, if the volume changes by some amount, the linear measures will always change by the cubic root of that amount. In this case, the volume of the sphere changed to be 1/3 of its original, so the linear change is the cubic root of 1/3. Multiplying 10 by the cubic root of 1/3 gives 6.9 which is almost 7. Our answer checks.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
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