answer:Let's denote the cost of the first 30 kgs of apples as ( l ) rupees per kilogram, the cost of each additional kilogram between 31-60 kgs as ( m ) rupees per kilogram, and the cost of each additional kilogram above 60 kgs as ( n ) rupees per kilogram. Given the price of 33 kilograms is 333 rupees, we can write the following equation: [ 30l + 3m = 333 ] (1) Given the price of 36 kilograms is 366 rupees, we can write the following equation: [ 30l + 6m = 366 ] (2) Given the price of 45 kilograms is 465 rupees, we can write the following equation: [ 30l + 15m = 465 ] (3) Given the price of 50 kilograms is 525 rupees, we can write the following equation: [ 30l + 20m = 525 ] (4) We can solve these equations to find the values of ( l ) and ( m ). From equation (1) and equation (2), we can subtract equation (1) from equation (2) to eliminate ( l ): [ 6m - 3m = 366 - 333 ] [ 3m = 33 ] [ m = 11 ] (5) Now that we have the value of ( m ), we can substitute it back into equation (1) to find ( l ): [ 30l + 3(11) = 333 ] [ 30l + 33 = 333 ] [ 30l = 333 - 33 ] [ 30l = 300 ] [ l = 10 ] (6) Now we have the cost of the first 30 kgs of apples as 10 rupees per kilogram. To find the cost of the first 20 kgs, we simply multiply 20 by the cost per kilogram ( l ): [ text{Cost of first 20 kgs} = 20 times l ] [ text{Cost of first 20 kgs} = 20 times 10 ] [ text{Cost of first 20 kgs} = 200 ] Therefore, the cost of the first 20 kgs of apples is boxed{200} rupees.

answer:According to the problem description, if the time required for the robot to run 50 meters with a stride length of x meters (allowing a distance exceeding 50 meters) is f(x) seconds, we can calculate the required steps and time to reach or exceed 50 meters for both given stride lengths. - For a stride length of x = 1.6 meters, the nearest number of steps to reach 50 meters without falling short is n = lceil frac{50}{1.6} rceil = 32. Note that taking 31 steps would cover 49.6 meters, which is less than 50. Therefore, the robot must take 32 full steps, and with each step taking 1.6 seconds, we have f(1.6) = 32 times 1.6. - For a stride length of x = 0.5 meters, the nearest number of steps to reach 50 meters without falling short is n = lceil frac{50}{0.5} rceil = 100. Note that taking 99 steps would cover only 49.5 meters, which is less than 50. Thus, the robot must take 100 full steps, and with each step taking 0.5 seconds, we have f(0.5) = 100 times 0.5. Now, let's calculate the actual values and the difference: f(1.6) = 32 times 1.6 = 51.2 text{ seconds}, f(0.5) = 100 times 0.5 = 50.0 text{ seconds}. Subtracting the two, we get: f(1.6) - f(0.5) = 51.2 - 50.0 = boxed{1.2} text{ seconds}. Thus, the correct answer is option B.

answer:To find the sides and diagonals of the quadrilateral formed by the intersection of the angle bisectors of the internal angles of a parallelogram with sides (a) and (b) and an angle (alpha), we proceed as follows: 1. **Identify the quadrilateral type**: The quadrilateral formed by the intersection of the angle bisectors of a parallelogram is known to be a rectangle. 2. **Determine the lengths of the sides**: Let's consider the properties of the angle bisectors in the context of the parallelogram. The angle bisectors will divide each angle into two equal parts. We need to determine the new lengths created by the intersection points of these bisectors. - Each angle of the parallelogram is split in half, so the angles in the resulting rectangle are (frac{alpha}{2}) and (frac{pi - alpha}{2}). 3. **Use trigonometric identities**: To find the sides of the resulting rectangle, consider the following: - Given the parallelogram sides (a) and (b), the sides of the rectangle can be found via trigonometric ratios involving (sin) and (cos) of (frac{alpha}{2}). Suppose the sides of the rectangle formed are denoted as (d_1) and (d_2). The equations for the sides can be derived from the relationships in a rectangle: - The side (d_1 = |a - b| cos frac{alpha}{2}) corresponds to how the lengths project when bisected and intersected in the rectangle. - The side (d_2 = |a - b| sin frac{alpha}{2}) similarly corresponds to another projection based on the bisection. 4. **Analyze the diagonal**: The distance between opposite corners of the rectangle (diagonal) is found using the Pythagorean theorem: [ text{Diagonal} = sqrt{d_1^2 + d_2^2} = sqrt{(|a - b| cos frac{alpha}{2})^2 + (|a - b| sin frac{alpha}{2})^2} ] Since (cos^2 frac{alpha}{2} + sin^2 frac{alpha}{2} = 1), we get: [ text{Diagonal} = |a - b| ] 5. **Conclusion:** Thus, the sides of the resulting quadrilateral rectangle are: [ d_1 = |a - b| cos frac{alpha}{2} ] [ d_2 = |a - b| sin frac{alpha}{2} ] And the diagonal is: [ text{Diagonal} = |a - b| ] Therefore, the lengths of the sides and the diagonal of the quadrilateral formed by the intersection of the angle bisectors of the parallelogram are [ |a - b| sin frac{alpha}{2}, quad |a - b| cos frac{alpha}{2}, quad text{and} quad |a - b| ] respectively. [boxed{|a - b| sin frac{alpha}{2}, |a - b| cos frac{alpha}{2}, |a - b|}]

answer:First, let's find out Sydney's age. We know that Randolph is 5 years older than Sydney, and Randolph is 55 years old. So, Sydney's age would be: Sydney's age = Randolph's age - 5 years Sydney's age = 55 years - 5 years Sydney's age = 50 years Now we know Sydney is 50 years old and Sherry is 25 years old. To find the ratio of Sydney's age to Sherry's age, we divide Sydney's age by Sherry's age: Ratio = Sydney's age / Sherry's age Ratio = 50 years / 25 years Ratio = 2/1 So, the ratio of Sydney's age to Sherry's age is boxed{2:1} .