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question:What is the perimeter, in units, of the triangle formed by the points (1, 3), (4, 6), and the origin (0, 0)?
answer:To find the perimeter, we need to calculate the distances between all pairs of points: between (1, -3) and (4, 6), between (1, -3) and (0, 0), and between (4, 6) and (0, 0). 1. **Distance between (1, -3) and (4, 6)**: [ sqrt{(4-1)^2 + (6-(-3))^2} = sqrt{3^2 + 9^2} = sqrt{9 + 81} = sqrt{90} ] 2. **Distance between (1, -3) and (0, 0)**: [ sqrt{(1-0)^2 + (-3-0)^2} = sqrt{1^2 + 3^2} = sqrt{1 + 9} = sqrt{10} ] 3. **Distance between (4, 6) and (0, 0)**: [ sqrt{(4-0)^2 + (6-0)^2} = sqrt{4^2 + 6^2} = sqrt{16 + 36} = sqrt{52} ] Add these distances to find the perimeter of the triangle: [ sqrt{90} + sqrt{10} + sqrt{52} ] The final perimeter is then boxed{sqrt{90} + sqrt{10} + sqrt{52}}.
question:f x + y = 250 and x y equals a certain number, for integers of x and y, y equals 225. What is the difference between x and y?
answer:Given that x + y = 250 and y = 225, we can substitute the value of y into the first equation to find x. x + 225 = 250 x = 250 - 225 x = 25 Now that we have the values of x and y, we can find the difference between x and y. Difference = x - y Difference = 25 - 225 Difference = -200 However, since the difference is typically expressed as a positive number, we take the absolute value. |Difference| = |-200| |Difference| = 200 So, the difference between x and y is boxed{200} .
question:Solve for x: frac{x+8}{x4} = frac{x3}{x+6}
answer:1. Cross-multiplying gives: [ (x+8)(x+6) = (x-3)(x-4) ] 2. Expanding both sides: [ x^2 + 6x + 8x + 48 = x^2 - 4x - 3x + 12 ] 3. Simplifying the equation: [ x^2 + 14x + 48 = x^2 - 7x + 12 ] 4. Combining like terms: [ 21x + 48 = 12 ] 5. Solving for x: [ 21x = 12 - 48 rightarrow 21x = -36 rightarrow x = frac{-36}{21} = frac{-12}{7} ] The solution is: [ x = boxed{frac{-12}{7}} ]
question:Angelo and Melanie want to plan how many hours over the next week they should study together for their test next week. They have 2 chapters of their textbook to study and 4 worksheets to memorize. They figure out that they should dedicate 3 hours to each chapter of their textbook and 1.5 hours for each worksheet. If they plan to study no more than 4 hours each day, how many days should they plan to study total over the next week if they take a 10minute break every hour, include 3 10minute snack breaks each day, and 30 minutes for lunch each day?
answer:To calculate the total study time Angelo and Melanie need, we start by considering the time they plan to dedicate to each component of their study material: - For the textbook chapters: They plan to study 3 hours for each of the 2 chapters. Therefore, the total time for chapters is: [3 , text{hours/chapter} times 2 , text{chapters} = 6 , text{hours}] - For the worksheets: They plan to dedicate 1.5 hours for each of the 4 worksheets. Thus, the total time for worksheets is: [1.5 , text{hours/worksheet} times 4 , text{worksheets} = 6 , text{hours}] Adding these together, the initial total study time is: [6 , text{hours (chapters)} + 6 , text{hours (worksheets)} = 12 , text{hours}] Next, we calculate the additional time needed for breaks and lunch: - For hourly breaks: They take a 10-minute break every hour for the 12 hours of study, which adds up to: [12 , text{hours} times 10 , text{minutes/hour} = 120 , text{minutes}] - For snack breaks: They take 3 snack breaks a day, each 10 minutes long: [3 , text{snack breaks} times 10 , text{minutes/break} = 30 , text{minutes}] - For lunch: They take 30 minutes for lunch each day. Adding these breaks together gives: [120 , text{minutes (hourly breaks)} + 30 , text{minutes (snack breaks)} + 30 , text{minutes (lunch)} = 180 , text{minutes}] Converting the total break time into hours: [180 , text{minutes} div 60 , text{minutes/hour} = 3 , text{hours}] Therefore, the total time needed, including study and breaks, is: [12 , text{hours (study)} + 3 , text{hours (breaks)} = 15 , text{hours}] Given they plan to study no more than 4 hours each day, the number of days they need to study is: [15 , text{hours} div 4 , text{hours/day} = 3.75] Since they cannot study for a fraction of a day, they will need to round up to the nearest whole number, which means they need to plan to study for (boxed{4}) days to accommodate all the time they need.