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question:Let D = a^2 + b^2 + c^2, where a and b are consecutive even integers, and c = a+b. Determine the nature of sqrt{D}: A. Always an even integer B. Sometimes an odd integer, sometimes not C. Always an odd integer D. Sometimes rational, sometimes not E. Always irrational
answer:1. **Define the variables**: Let a = 2x and b = 2x + 2 (consecutive even integers). Then, c = 2x + 2x + 2 = 4x + 2. [ D = a^2 + b^2 + c^2 = (2x)^2 + (2x+2)^2 + (4x+2)^2 ] 2. **Simplify the expression for D**: [ D = 4x^2 + (4x^2 + 8x + 4) + (16x^2 + 32x + 4) = 24x^2 + 40x + 8 ] 3. **Check if D is a perfect square**: We'd be checking if 24x^2 + 40x + 8 can be expressed as (Ax^2 + Bx + C)^2. Simplifying not likely yields a perfect square directly, so we aim to find sqrt{D} directly. [ sqrt{D} = sqrt{24x^2 + 40x + 8} = sqrt{8(3x^2 + 5x + 1)} ] The expression within the square root is not a perfect square polynomial unless specific values of x are chosen. However, in general cases, D under the square root would yield an irrational result due to the presence of a non-square factor. 4. **Conclusion**: From the formation of D and evaluating the square root, it's evident that: [ textbf{(E) text{always irrational}} ] The final answer is boxed{E. Always irrational}
question:A3B41=591. How much is A?
answer:To find the value of A, we need to solve the equation A3B - 41 = 591. First, let's add 41 to both sides of the equation to isolate the term with A on one side: A3B + 41 - 41 = 591 + 41 A3B = 632 Now, we need to determine what A could be to make the number A3B a three-digit number that when subtracted by 41 equals 591. Since we are dealing with whole numbers, A must be a digit from 0 to 9. Let's try to find A by dividing 632 by 100 (since A is in the hundreds place): 632 ÷ 100 = 6.32 Since A must be a whole number, we can see that A = 6 because 6 is the whole number part of the quotient. Therefore, A = boxed{6} .
question:Omi is twice as old as Kimiko. Arlette is 3/4 times as old as Kimiko. If Kimiko is 28 years old, calculate the average age of the three?
answer:First, let's find the ages of Omi and Arlette based on Kimiko's age. Omi's age: Omi is twice as old as Kimiko, so we multiply Kimiko's age by 2. Omi's age = 2 * Kimiko's age Omi's age = 2 * 28 Omi's age = 56 years old Arlette's age: Arlette is 3/4 times as old as Kimiko, so we multiply Kimiko's age by 3/4. Arlette's age = (3/4) * Kimiko's age Arlette's age = (3/4) * 28 Arlette's age = 21 years old Now, let's calculate the average age of the three. Average age = (Omi's age + Kimiko's age + Arlette's age) / 3 Average age = (56 + 28 + 21) / 3 Average age = 105 / 3 Average age = 35 years old The average age of the three is boxed{35} years old.
question:John bought 9.25 m of cloth. The cost price per metre is 48. How much did John pay for the cloth in total?
answer:To find the total cost that John paid for the cloth, we need to multiply the cost per metre by the total metres of cloth he bought. Total cost = Cost per metre × Total metres of cloth Total cost = 48 × 9.25 m Now, let's calculate the total cost: Total cost = 48 × 9.25 Total cost = 444 So, John paid boxed{444} for the cloth in total.