2019 Mock AMC 10B Problems/Problem 18

we can first draw the initial position and the final position. As the smaller log does not move, we can combine the two figures into one larger figure with both larger logs. During that time the log is rolling, we can model it as a part of a circle. That circle's radius is 1+3+3, which is the sum of the smaller log's radius and the larger log's diameter. We need to find the volume, so that means we need to find the angle of that portion of the semicircle. We can find the angle by drawing a drawing of the altitudes from the center to the ground. After that, we can connect the two centers and find out that it is a triangle. This triangle has a smaller length of 2 and a hypotenuse of 4. This means that the angle is 30 degrees. We can do the same for the other side, and we find out that the angle of the portion of the semicircle is 180-30-30, which is 120. Now, we can find the area. The area of that portion is 49/3 pi, and the combined area of the initial and final position gives us a total of 76/3 pi. However, we are finding the volume encompassed by the larger log rolling over the smaller log, and we have to subtract 1/3 pi from the total area. Multiplying by 10 to find the volume, we get a final result of 750/3, or 250 pi. This means that out answer is (A)