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question:Let (a,b,c,d) be an ordered quadruple of not necessarily distinct integers, each one of them in the set {1,2,3,4}. For how many such quadruples is it true that acdot d bcdot c is odd? A) 64 B) 72 C) 96 D) 120 E) 144
answer:To find the number of ordered quadruples (a, b, c, d) such that acdot d - bcdot c is odd, we again use a parity-based approach. Step 1: Understanding Parity of Products for New Set - a cdot d is odd if both a and d are odd. - b cdot c is odd if both b and c are odd. - In the set {1, 2, 3, 4}, the odd numbers are 1 and 3. Step 2: Conditions for acdot d - bcdot c to be Odd 1. a cdot d is odd and b cdot c is even. 2. a cdot d is even and b cdot c is odd. Step 3: Counting Suitable Quadruples - **Case 1**: Odd a cdot d, Even b cdot c - Odd choices for a, d: (1, 3) providing 2 times 2 = 4 ways. - Even choices for b cdot c: Complement within 4 times 4 = 16 total pairings minus the 4 ways both are odd gives 16 - 4 = 12 ways. - **Case 2**: Even a cdot d, Odd b cdot c - By symmetry, this case also provides 4 times 12 = 48 ways. Step 4: Total Quadruples Total 48 + 48 = 96 quadruples. Conclusion The total number of ordered quadruples (a, b, c, d) such that acdot d - bcdot c is odd in the set {1, 2, 3, 4} is 96. The final answer is boxed{C}
question:The interior of a region is bounded by the graphs of (x+ay)^3 = 8a^3 and (axy)^2 = 4a^2, where a is a positive real number. What is the area of this region in terms of a, valid for all a > 0? A) frac{8a^2}{(a+1)^2} B) frac{16a^2}{a^2+1} C) frac{8a}{a^2+1} D) frac{8a}{a+1} E) frac{4a}{a+1}
answer:Step 1: Analyze and rewrite the given equations 1. ((x+ay)^3 = 8a^3) 2. ((ax-y)^2 = 4a^2) These can be rewritten as: - For equation 1: (x+ay = 2a) (not considering complex roots) - For equation 2: (ax-y = pm 2a) Step 2: Convert to standard line equations From the rewritten forms: - (x + ay = 2a) - (ax - y = 2a) and (ax - y = -2a) Rewritten: - (y = -frac{x}{a} + 2) - (y = ax - 2a) and (y = ax + 2a) Step 3: Calculate intersections of lines and resulting figure The intersecting points of lines (y = -frac{x}{a} + 2) with (y = ax - 2a) and (y = ax + 2a) need to be found. Intersection with (y = ax - 2a): [ -frac{x}{a} + 2 = ax - 2a ] [ x(a + frac{1}{a}) = 2 + 2a ] [ x = frac{2(1+a)}{a + frac{1}{a}} ] Same calculation follows for intersection with (y = ax + 2a), establishing vertices of a triangle with these and vertex at origin. Step 4: Calculate the area of the triangle Use the formula for area of triangle with vertices ( (0,0), (x_1, y_1), (x_2, y_2) ): [ text{Area} = frac{1}{2} left| x_1y_2 - x_2y_1 right| ] Substitute the coordinates after computing their specific values using (x) and respective (y) values. Conclusion: Calculating specific area and expressing it in terms of (a), [ text{Area} = frac{8a{a^2+1}} ] The correct answer is C) boxed{frac{8a}{a^2+1}}.
question:’s father has an egg business supplying the local market. On a Wednesday morning, his father sends him to go and collect 10 trays of eggs for sale from their store. While loading the eggs into the car, he accidentally drops two trays. He calls his father telling him this, and is told to add 7 more trays for sale. How many eggs were sold that day?
answer:First, let's determine how many eggs were initially supposed to be sold before the accident: 1 tray of eggs typically contains 30 eggs. Haman was supposed to collect 10 trays of eggs. So, initially, 10 trays x 30 eggs/tray = 300 eggs were supposed to be sold. However, Haman accidentally drops 2 trays, so those are not sold: 2 trays x 30 eggs/tray = 60 eggs were lost. Subtracting the lost eggs from the initial amount gives us: 300 eggs - 60 eggs = 240 eggs remaining. After the accident, his father tells him to add 7 more trays for sale: 7 trays x 30 eggs/tray = 210 eggs to be added. Adding the additional eggs to the remaining eggs gives us: 240 eggs + 210 eggs = 450 eggs. Therefore, boxed{450} eggs were sold that day.
question:A rectangle and a regular hexagon are designed such that the area of the rectangle is numerically equal to twice its perimeter, and the area of the hexagon is numerically equal to three times its perimeter. If the length of the rectangle is twice its width, compare the apothem of the rectangle (which is half its width) with the apothem of the hexagon. A) frac{1}{3}text{ times the second} B) frac{1}{2}text{ times the second} C) 1text{ time the second} D) 2text{ times the second} E) 3text{ times the second}
answer:1. **Determine the dimensions of the rectangle:** Let the width of the rectangle be w and the length be 2w. The area of the rectangle is 2w times w = 2w^2 and its perimeter is 2(w + 2w) = 6w. Given that the area is twice the perimeter: [ 2w^2 = 2 times 6w implies w^2 = 6w implies w(w - 6) = 0 ] We discard w = 0, so w = 6 and the length is 2w = 12. 2. **Calculate the apothem of the rectangle:** [ text{Apothem of rectangle} = frac{w}{2} = frac{6}{2} = 3 ] 3. **Determine the side length of the hexagon:** Let the side length of the hexagon be h. The area of a regular hexagon is frac{3sqrt{3}}{2} h^2 and its perimeter is 6h. Given that the area is three times the perimeter: [ frac{3sqrt{3}}{2} h^2 = 3 times 6h implies frac{3sqrt{3}}{2} h^2 = 18h implies sqrt{3} h = 12 implies h = frac{12}{sqrt{3}} = 4sqrt{3} ] 4. **Calculate the apothem of the hexagon:** The apothem of a regular hexagon is also the radius of the inscribed circle, which is: [ frac{sqrt{3}}{2} h = frac{sqrt{3}}{2} times 4sqrt{3} = 6 ] 5. **Compare the apothems:** The apothem of the rectangle is 3 and the apothem of the hexagon is 6. Conclusion: The apothem of the hexagon is twice the apothem of the rectangle. Therefore, the answer is textbf{B frac{1}{2}text{ times the second}}. The final answer is boxed{textbf{(B)} frac{1}{2}text{ times the second}}