answer:To solve this problem, let's examine each option step by step: **Option A: 3+sqrt{3}** This expression cannot be simplified to 3sqrt{3} because the terms are not like terms. The correct simplification would just leave the expression as it is, since 3 and sqrt{3} cannot be combined through addition. **Option B: 2sqrt{3}+sqrt{3}** This expression can be simplified by combining like terms. We have two terms that are multiples of sqrt{3}, so we add the coefficients (2 and 1) together: [2sqrt{3}+sqrt{3} = (2+1)sqrt{3} = 3sqrt{3}] This matches the given expression, so this option is correct. **Option C: 2sqrt{3}-sqrt{3}** Similar to option B, we combine like terms by subtracting the coefficients: [2sqrt{3}-sqrt{3} = (2-1)sqrt{3} = sqrt{3}] This does not simplify to 2, so this option is not correct. **Option D: sqrt{3}+sqrt{2}** These terms cannot be combined through addition because they involve square roots of different numbers. There is no simplification that leads to sqrt{5}, so this option is not correct. Given the analysis above, the correct answer is: [boxed{B}]

answer:This problem tests our understanding of the properties of arithmetic sequences, specifically the sum of their first n terms. The key lies in understanding and applying these properties. Since the given sequence is an arithmetic sequence, the sum of its first 10 terms, the sum of its second 10 terms, and so on, also form an arithmetic sequence. Using S_{10}=10 and S_{20}=30, we can find the value of S_{30}. Here's the step-by-step solution: Since {a_n} is an arithmetic sequence, S_{10}, S_{20}-S_{10}, and S_{30}-S_{20} also form an arithmetic sequence. Given S_{10}=10 and S_{20}=30, We have 2 times 20 = 10 + (S_{30}-30), Solving for S_{30}, we get boxed{S_{30}=60}.

answer:The points (0,0), (0,4), (x,4), and (x,0) form a rectangle with one side along the y-axis and the other side parallel to the x-axis. The line x + y = 4 is a straight line that passes through the points (0,4) and (4,0) in the x-y plane. The area below this line and above the x-axis in the first quadrant is a right-angled triangle with vertices at (0,0), (0,4), and (4,0). The probability that a randomly chosen point in the rectangle lies below the line x + y < 4 is given as 0.4. This means that the area of the triangle formed by the line x + y = 4 and the x-axis is 0.4 times the area of the rectangle. The area of the rectangle is the product of its length and width. Since one side is along the y-axis from (0,0) to (0,4), the width is 4. The length is the distance along the x-axis from (0,0) to (x,0), which is x. So, the area of the rectangle is 4x. The area of the right-angled triangle is 1/2 * base * height. The base and height of the triangle are both 4 (from (0,0) to (4,0) and from (0,0) to (0,4)), so the area of the triangle is 1/2 * 4 * 4 = 8. Since the area of the triangle is 0.4 times the area of the rectangle, we can set up the following equation: Area of triangle = 0.4 * Area of rectangle 8 = 0.4 * 4x 8 = 1.6x Now, we solve for x: x = 8 / 1.6 x = 5 Therefore, the x-coordinate of the other two points of the rectangle is boxed{5} .

answer:To solve this problem, we start by understanding the relationship between the base lengths and the legs of the isosceles triangles in the two scenarios given. The first scenario has a base length of 4.848 millimeters and legs of 1 kilometer each, with a vertex angle of 1″. The second scenario asks us to find the base length when the legs are the average distance from the sun to the earth, which is 1.5times 10^{8} kilometers, and the vertex angle remains 1″. Given that the relationship between the base lengths and the legs should be proportional when the vertex angle is constant, we set up a proportion based on the information provided: [ frac{1}{4.848} = frac{1.5times 10^{8}}{x} ] Here, x represents the base length in millimeters when the legs are 1.5times 10^{8} kilometers long. Solving for x, we multiply both sides by x and then by 4.848 to isolate x: [ x = 1.5times 10^{8} times 4.848 ] Calculating the right side gives us: [ x = 7.272times 10^{8} text{ millimeters} ] To convert millimeters to kilometers, we remember that 1 kilometer equals 10^{6} millimeters. Therefore, [ 7.272times 10^{8} text{ millimeters} = 727.2 text{ kilometers} ] Thus, the base length of the isosceles triangle with a vertex angle of 1″, when the legs are the average distance from the sun to the earth, is 727.2 kilometers. Therefore, the correct answer is boxed{D}.