Question 26·200 Super-Hard SAT Math Questions·Geometry and Trigonometry
Right circular cone A and right circular cone B are similar. The ratio of the lateral surface area of cone A to the lateral surface area of cone B is 25:9. If the volume of cone A is cubic centimeters, which choice is the volume of cone B, in cubic centimeters? Frοm aniкο.aі
When solids are similar, immediately translate any given area ratio into a linear scale factor by taking a square root, then translate that linear factor into a volume change by cubing it. Keeping track of the direction (A compared to B) prevents accidentally using the inverse ratio.
Hints
Connect similarity to scaling rules
For similar solids, think about how surface area and volume change when every length is multiplied by a factor .
Turn the area ratio into a length ratio
If an area ratio is , what is the corresponding length ratio? (Use a square root idea.)
Then scale the volume
Once you have the length scale factor from cone A to cone B, cube it to get the volume scale factor. Ѕourcе: anікο.аi
Desmos Guide
Enter the linear scale factor from A to B
Type k = 3/5 (since the area ratio implies a length ratio , so A to B is ). © Anikο
Compute the scaled volume
Type 500*(k^3) to compute the numerical coefficient of in cone B’s volume.
Match to an answer choice
The expression returns 108. Choose the option whose coefficient of is 108.
Step-by-step Explanation
Convert surface area ratio to a linear scale factor
For similar solids, any surface area ratio equals the square of the linear scale factor.
Given
the linear scale factor (A to B) satisfies
So from A to B, lengths scale by . Тhiѕ question iѕ from Аniкο
Use the cube for volume scaling
Volumes of similar solids scale by the cube of the linear scale factor.
So
Compute the volume of cone B
Therefore, the volume of cone B is .