answer:Let's denote the sides of the rectangle as length ( L ) and width ( W ). The area of the rectangle (( A_{rectangle} )) is given by: [ A_{rectangle} = L times W ] Now, let's consider the triangle. If one of the sides of the rectangle is used as the base of the triangle, then the base of the triangle is either ( L ) or ( W ). Without loss of generality, let's assume the base of the triangle is ( L ). The height of the triangle would then be ( W ), since the vertex of the triangle is on the opposite side of the rectangle. The area of the triangle (( A_{triangle} )) is given by: [ A_{triangle} = frac{1}{2} times text{base} times text{height} ] [ A_{triangle} = frac{1}{2} times L times W ] According to the problem, the ratio of the area of the rectangle to the area of the triangle is 2. So we have: [ frac{A_{rectangle}}{A_{triangle}} = 2 ] Substituting the expressions for ( A_{rectangle} ) and ( A_{triangle} ), we get: [ frac{L times W}{frac{1}{2} times L times W} = 2 ] Simplifying the equation: [ frac{L times W}{frac{1}{2}L times W} = frac{L times W}{frac{L times W}{2}} = frac{L times W}{1} times frac{2}{L times W} = 2 ] So the ratio is indeed boxed{2,} as given in the problem. This means that the area of the rectangle is twice the area of the triangle when one side of the rectangle is used as the base of the triangle and the vertex is on the opposite side.

answer:Given the parabola y=ax^{2}+bx+c where a, b, and c are constants with aneq 0 and c > 3, and it passes through the point (5,0) with its axis of symmetry being the line x=2, we analyze the given conclusions. 1. **Finding the other intersection point with the x-axis:** Since the axis of symmetry is x=2 and the parabola passes through (5,0), by symmetry, the other intersection point with the x-axis is at (2 - (5-2), 0) = (-1, 0). 2. **Determining the direction of the parabola:** Given c > 3, and knowing the parabola intersects the x-axis at two points, it must open downwards to satisfy both conditions. Thus, a < 0. 3. **Analyzing the sign of b:** The axis of symmetry formula is x = -frac{b}{2a}. Given the axis of symmetry is x=2, we have 2 = -frac{b}{2a}. Since a < 0, for the equation to hold, b must be positive. Thus, b > 0. 4. **Conclusion ①: abc < 0:** Since a < 0, b > 0, and c > 3, it follows that abc < 0. Therefore, conclusion ① is correct. 5. **Conclusion ②: Two distinct real roots for ax^{2}+bx+c=2:** The parabola intersects the x-axis at two points and opens downwards, with c > 3 indicating the vertex is above the line y=2. Therefore, the equation ax^{2}+bx+c=2 must have two distinct real roots. Conclusion ② is correct. 6. **Conclusion ③: a < -frac{3}{5}:** Since the parabola passes through (5,0), we have 25a+5b+c=0. With x=-frac{b}{2a}=2, we find b=-4a. Substituting b=-4a into 25a+5b+c=0 gives 5a+c=0, leading to c=-5a. Given c > 3, we have -5a > 3, which simplifies to a < -frac{3}{5}. Thus, conclusion ③ is correct. Since all conclusions ①, ②, and ③ are correct, the correct answer is: boxed{D}

answer:1. **Identify the expressions for ( x ) and ( y ):** [ x = cos 1^circ cos 2^circ cos 3^circ cdots cos 89^circ ] [ y = cos 2^circ cos 6^circ cos 10^circ cdots cos 86^circ ] 2. **Express ( y ) in terms of sine using the identity (cos theta = sin (90^circ - theta)):** [ y = cos 2^circ cos 6^circ cos 10^circ cdots cos 86^circ = sin 88^circ sin 84^circ sin 80^circ cdots sin 4^circ ] 3. **Use the sine double angle formula (sin 2theta = 2 sin theta cos theta) in reverse:** [ y = (2 sin 44^circ cos 44^circ)(2 sin 42^circ cos 42^circ) cdots (2 sin 2^circ cos 2^circ) ] [ y = 2^{22} sin 44^circ sin 42^circ cdots sin 2^circ cos 44^circ cos 42^circ cdots cos 2^circ ] 4. **Combine the sine terms:** [ y = 2^{22} sin 2^circ sin 4^circ cdots sin 88^circ ] 5. **Apply the double angle formula again:** [ y = 2^{22} cdot 2^{22} sin 1^circ sin 2^circ cdots sin 44^circ sin 46^circ sin 47^circ cdots sin 89^circ ] [ y = 2^{44} sin 1^circ sin 2^circ cdots sin 44^circ sin 46^circ sin 47^circ cdots sin 89^circ ] 6. **Use (sin theta = cos (90^circ - theta)) again:** [ y = frac{2^{44} cos 1^circ cos 2^circ cdots cos 89^circ}{sin 45^circ} ] [ sin 45^circ = frac{1}{sqrt{2}} ] [ y = 2^{44} cdot sqrt{2} cdot cos 1^circ cos 2^circ cdots cos 89^circ ] 7. **Divide ( y ) by ( x ):** [ frac{y}{x} = frac{2^{44} cdot sqrt{2} cdot cos 1^circ cos 2^circ cdots cos 89^circ}{cos 1^circ cos 2^circ cdots cos 89^circ} = 2^{44} cdot sqrt{2} ] [ frac{y}{x} = 2^{44} cdot 2^{1/2} = 2^{44.5} ] 8. **Calculate (frac{2}{7} log_2 frac{y}{x}):** [ frac{2}{7} log_2 (2^{44.5}) = frac{2}{7} cdot 44.5 = frac{89}{7} approx 12.714 ] 9. **Find the nearest integer:** [ text{The nearest integer to } 12.714 text{ is } 13. ] The final answer is (boxed{13})

answer:Given the equation x + y = 50, we observe that for each value of x from 1 to 49, there corresponds a unique value of y such that the sum is 50. To find this corresponding y, we use y = 50 - x. Now, we shall calculate the ordered pairs: - When x = 1, y = 50 - 1 = 49. So, we have the pair (1, 49). - When x = 2, y = 50 - 2 = 48. So, we have the pair (2, 48). - ... - When x = 49, y = 50 - 49 = 1. So, we have the pair (49, 1). Thus, as x ranges from 1 to 49, there are 49 distinct integers it can take, and for each integer value of x, there is a unique integer value of y. Hence, there are boxed{49} distinct ordered pairs (x, y).