Diffusion

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question:w many triangles with positive area are there whose vertices are points in the xy-plane whose coordinates are integers (x,y) satisfying 1le xle 4 and 1le yle 4? text{(A) } 496quad text{(B) } 500quad text{(C) } 512quad text{(D) } 516quad text{(E) } 560

answer:We can choose any three distinct points from the 16 lattice points in the given square. By the Shoelace Theorem, the area of a triangle with coordinates (x_1,y_1), (x_2,y_2), and (x_3,y_3) is [frac12 |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|.] The absolute value of this expression is minimized when the three points form a right triangle, and since we are only interested in triangles with positive area, the three points must be collinear or form a right triangle. There are 16 cdot 15 cdot 14 ways to choose three distinct points, but we have counted the collinear triangles three times each (once for each order of the points), and we have counted the right triangles twice each (once for each order of the two non-right-angle points). There are 16 cdot 15 cdot 14 - 16 cdot 15 cdot 3/2 - 16 cdot 3 cdot 2 = boxed{516} triangles. The answer is: 516

question:Fifi's closet, she hangs all of her clothes on colored plastic hangers. She has clothes hanging on x pink hangers, 4 green hangers, one less blue hanger than there are green hangers, and one less yellow hanger than there are blue hangers. What is the total number of colored hangers in Fifi's closet? If we know the answer to the above question is 16, what is the value of unknown variable x?

answer:Fifi has x pink hangers. She has 4 green hangers. She has one less blue hanger than there are green hangers, which means she has 4 - 1 = 3 blue hangers. She has one less yellow hanger than there are blue hangers, which means she has 3 - 1 = 2 yellow hangers. The total number of colored hangers in Fifi's closet is the sum of the number of pink, green, blue, and yellow hangers: x + 4 + 3 + 2 = x + 9. We are given that the total number of colored hangers is 16, so we can write: x + 9 = 16. Solving for x, we get: x = 7. The value of x is 7. 7 The answer is: 7

question:Round to the nearest tenth: 36.89753

answer:we look at the digit to the right, which is 3. Since 3 is less than 5, we round down and keep the tenths digit the same. Therefore, rounding 36.89753 to the nearest tenth gives us boxed{36.9}.The answer is: 36.9

question:Arnold owns x cars. The first car averages 50 miles per gallon of gas. The second car averages 10 miles per gallon of gas. And the third car averages 15 miles per gallon of gas. He splits his 450-mile monthly driving mileage equally amongst his three cars. If gas costs 2 per gallon, He spends 56 on gas each month. What is the value of unknown variable x? What is the value of unknown variable x?

answer:To solve this problem, we need to determine the value of x, which represents the number of cars Arnold owns. Let's break down the information given: Average miles per gallon for the first car: 50 Average miles per gallon for the second car: 10 Average miles per gallon for the third car: 15 Monthly driving mileage: 450 miles Gas cost per gallon: 2 Total monthly gas cost: 56 We can set up the equation as follows: (Monthly driving mileage / Average miles per gallon for the first car) + (Monthly driving mileage / Average miles per gallon for the second car) + (Monthly driving mileage / Average miles per gallon for the third car) = Total monthly gas cost / Gas cost per gallon (450 / 50) + (450 / 10) + (450 / 15) = 56 / 2 Let's simplify and solve for x: 9 + 45 + 30 = 28 84 = 28 To solve for x, we divide both sides of the equation by 28: x = 84 / 28 x = 3 The value of x is 3. Arnold owns 3 cars. 3 The answer is: 3