answer:The ceiling function rounds up to the nearest whole number. We have lceil sqrt{2} rceil = 2, lceil sqrt{22} rceil = 5, and lceil sqrt{222} rceil = 15. So the sum is 2+5+15 = boxed{22}. The answer is: 22

answer:The top level of the pyramid has just one case. The second level has a base of 2x2 cases, so it needs 2x2 = 4 cases. The third level has a base of 3x3 cases, so it needs 3x3 = 9 cases. The fourth level has a base of 4x4 cases, so it needs 4x4 = 16 cases. To find the total number of cases needed, we add up the number of cases for each level: 1 + 4 + 9 + 16 = 30 Mark needs 30 cases of soda to build the pyramid. 30 The answer is: 30

answer:The sum of the 12 scores is 12 times 82 = 984. The sum of the remaining 10 scores is 10 times 84 = 840. The difference between the sum of all 12 scores and the sum of the remaining 10 scores is 984 - 840 = 144. This difference is equal to the sum of the highest and lowest scores, so the lowest score is 144 - 98 = boxed{46}. The answer is: 46

answer:The area of a circle is given by the formula A=pi r^2, where r is the radius of the circle. We are given that the area is smaller than 60pi, so we have the inequality pi r^2 < 60pi. Dividing both sides of the inequality by pi, we get r^2 < 60. Taking the square root of both sides, we get r < sqrt{60}. Since we are looking for the largest possible integer value for the radius, we take the floor function of sqrt{60}, which is boxed{7}. The answer is: 7