Grades 11-12 Video Solutions 2025-2025_11-12

Video Transcription

disks with positive radii R1, R2, R3, and R4 are centered at 0,0, 1,0, 3,0, and 6,0. The disks may touch but not overlap. What is the largest possible value of R1 plus R2  plus R3 plus R4? And we are given answer choices of A, 3, B, 4, C, 5, D, 6, and E. There is no upper limit.  So we can start with maybe some inequalities and get some relationships between each thing. So we can assign this circle to have radius 1, this circle to have radius 2, this circle  to have radius 3, and this circle to have radius 4. Note that we are not making any assumptions, we're just labeling our variables. Then we know that if the radii are tangent  to one another, that means that this entire line segment is filled up by R1 plus R2. And we know that R1 plus R2 can't be greater than 1 because that would imply that these circles  overlap. So we have that R1 plus R2 is actually less than or equal to 1. By very similar logic,  we get that R2 plus R3 is less than or equal to 2. And then finally, over here, R3 plus R4 is less than or equal to 3. But if you notice, we're just looking for this sum here.  And there are two component pieces here, two inequalities, that we can put together to get R1 plus R2 plus R3 plus R4. And that's, yeah, like these two inequalities. And we  get that like summing those together, we get that it's less than or equal to 4. So immediately  we know that C, D, and E are not valid answer choices. And now we just need to actually construct some shapes that actually work. And we can verify, if you'd like, that R1  equals 0.5, R2 equals 0.5, R3 equals 1.5, and R4 equals 1.5. We can check that that suffices. I'll leave that as an exercise to you, though. So our answer should be 4,  because the sum of all of these things, you know, 0.5, 1, 2.5, 4. So our answer is B.

Video Summary

The problem involves finding the maximum sum of radii \( R1 + R2 + R3 + R4 \) for disks centered at specified points such that they touch but do not overlap. By establishing inequalities (e.g., \( R1 + R2 \leq 1 \), \( R2 + R3 \leq 2 \), \( R3 + R4 \leq 3 \)), it is deduced that the combined radius \( R1 + R2 + R3 + R4 \) must be \( \leq 4 \). The answer choices C, D, and E exceed this maximum, leaving B (4) as the only valid solution. A combination such as \( R1 = 0.5, R2 = 0.5, R3 = 1.5, R4 = 1.5 \) satisfies the conditions. Thus, the answer is B.