answer:# Problem: Given a right-angled triangle ABC with legs AC = a and BC = b. Find: a) The side length of the square (with vertex C) that has the largest area entirely within triangle ABC. b) The dimensions of the rectangle (with vertex C) that has the largest area entirely within triangle ABC. Let's solve each part step-by-step. Part (a): 1. **Determine the Side Length of the Square**: * We'll start by noting that the square with maximum area within triangle ABC with vertex C will have equal side lengths s such that the vertices of the square lie along AC and BC. 2. **Use Similar Triangles**: * Consider the largest such square inside the triangle with sides parallel to the legs AC and BC. Notice that, by similarity of triangles, the smaller triangles outside the square and the overall triangle related proportionally. 3. **Set up the Proportional Relations**: * These relations show that s = frac{ab}{a+b} - derived from the algebraic manipulation, usually by considering similar triangles in {triangle ABC}. Hence, the side length of the square is: [ s = frac{ab}{a+b} ] So for part (a), the side length of the largest possible square is: [ boxed{frac{ab}{a+b}} ] Part (b): 1. **Let the Dimensions of the Rectangle be x and y**: - Here, we need to find x and y such that xy is maximized given x and y are sides of the rectangle within triangle ABC with vertex C. 2. **Apply the Constraint Relation**: - Place a coordinate system with C at the origin. Then, the equation of the hypotenuse line of triangle ABC is: [ frac{x}{a} + frac{y}{b} = 1 ] 3. **Express y in Terms of x and Then Find Area Formula**: - From the line equation, isolate one variable in terms of the other: [ y = b left( 1 - frac{x}{a} right) ] - The area (S) of the rectangle is then: [ S = x cdot y = x cdot b left(1 - frac{x}{a} right) = bx - frac{b}{a}x^2 ] 4. **Find the Maximum Area**: - This can be solved by differentiation: [ frac{dS}{dx} = b - frac{2b}{a}x = 0 ] - Solving for x: [ b - frac{2b}{a}x = 0 implies x = frac{a}{2} ] - Substitute x = frac{a}{2} back into the constraint equation to find y: [ y = b left(1 - frac{frac{a}{2}}{a} right) = b left(1 - frac{1}{2} right) = frac{b}{2} ] Thus, the dimensions (x) and (y) are: [ boxed{frac{a}{2} text{ and } frac{b}{2}} ]

answer:Since the inequality 0 leqslant x^{2}-ax+a leqslant 1 has only one unique solution, we can infer that the equation x^{2}-ax+a=1 also has only one unique solution. This implies that the discriminant Delta = a^{2} - 4(a-1) = 0. Solving this equation gives us a^{2} - 4a + 4 = 0, which has a solution a = 2. Therefore, the answer is boxed{2}. This solution is derived by utilizing the properties of quadratic functions, transforming the given inequality into an equation to find the unique solution. This problem primarily tests the understanding of the relationship between quadratic functions and inequalities, making it a fundamental question.

answer:To find out how much Maya's organization made from cauliflower sales, we can follow these steps: 1. Calculate the sales of carrots, knowing that they are twice the sales of broccoli: - Sales of carrots = 57 times 2 = 114. 2. Determine half the sales of carrots to find the base for calculating spinach sales: - Half the sales of carrots = frac{114}{2} = 57. 3. Calculate the sales of spinach, which is 16 more than half of the sales of carrots: - Sales of spinach = 57 + 16 = 73. 4. Add up the total sales of broccolis, carrots, and spinach to find out the combined sales before calculating the cauliflower sales: - Total sales of broccolis, carrots, spinach = 57 + 114 + 73 = 244. 5. Subtract the total sales of broccolis, carrots, and spinach from the overall earnings to find the sales from cauliflower: - Sales from cauliflower = 380 - 244 = boxed{136}. Therefore, Maya's organization made boxed{136} from cauliflower sales.

answer:Let's denote the number of votes the winner initially received as W and the number of votes the loser initially received as L. The total number of votes polled is 15000, so we have: W + L = 15000 If 3000 people change their mind and vote for the loser instead, the loser's votes become L + 3000 and the winner's votes become W - 3000. According to the problem, the loser would then win by a margin of 20% of the votes polled. This means that the loser would have 60% of the total votes and the winner would have 40% of the total votes, because the margin is the difference between the two percentages (60% - 40% = 20%). So we have: L + 3000 = 0.6 * 15000 W - 3000 = 0.4 * 15000 Now we can solve for L and W: L + 3000 = 9000 L = 9000 - 3000 L = 6000 W - 3000 = 6000 W = 6000 + 3000 W = 9000 Now we know the initial number of votes for the winner (W) and the loser (L): W = 9000 L = 6000 The initial winning margin is the difference between the winner's and loser's votes: Initial Margin = W - L Initial Margin = 9000 - 6000 Initial Margin = 3000 To find the initial winning margin percentage, we divide the initial margin by the total number of votes polled and multiply by 100: Initial Margin Percentage = (Initial Margin / Total Votes Polled) * 100 Initial Margin Percentage = (3000 / 15000) * 100 Initial Margin Percentage = 0.2 * 100 Initial Margin Percentage = 20% So the initial winning margin percentage was also boxed{20%} .