answer:To find the starting number, we need to find the range of numbers that includes exactly 12 numbers divisible by 3, ending with 47. Since 47 is not divisible by 3, we need to find the largest number less than or equal to 47 that is divisible by 3. We can do this by dividing 47 by 3 and taking the integer part of the result: 47 ÷ 3 = 15 with a remainder of 2 So the largest multiple of 3 less than or equal to 47 is 15 * 3 = 45. Now we need to find the smallest number in the range that gives us exactly 12 numbers divisible by 3. Since each consecutive multiple of 3 is 3 units apart, we can find the starting number by subtracting 11 multiples of 3 from 45 (we subtract 11 because we are including 45 as one of the 12 numbers): Starting number = 45 - (11 * 3) = 45 - 33 = 12 Therefore, the starting number is boxed{12} .

answer:1. **Use Pythagorean Theorem in triangle ABC:** Let AB = x and BC = y. Since angle ABC = 90^circ, by the Pythagorean theorem, we have: [ x^2 + y^2 = AC^2 = 25^2 = 625. ] 2. **Calculate Area of triangle ACD:** Since angle ACD = 90^circ, the area of triangle ACD is: [ [ACD] = frac{1}{2} cdot AC cdot CD = frac{1}{2} cdot 25 cdot 40 = 500. ] 3. **Use Similar Triangles to Determine Segments EF and BF:** [ EC = AC - AE = 25 - 8 = 17. ] Since triangle CEF sim triangle CAB, based on AA similarity (angle CEF = angle CAB = 90^circ and angle ECF = angle BCA): [ frac{EF}{AB} = frac{CE}{CA} implies EF = AB cdot frac{CE}{CA} = x cdot frac{17}{25}. ] Similarly, [ frac{CF}{BC} = frac{CE}{CA} implies CF = BC cdot frac{17}{25} = y cdot frac{17}{25}. ] Thus, [ BF = BC - CF = y - frac{17y}{25} = frac{8y}{25}. ] 4. **Using Pythagorean Theorem in triangle BEF:** [ BE = sqrt{EF^2 + BF^2} = sqrt{left(frac{17x}{25}right)^2 + left(frac{8y}{25}right)^2} = frac{sqrt{289x^2 + 64y^2}}{25}. ] Substituting x^2 + y^2 = 625, we have: [ BE = frac{sqrt{289x^2 + 64(625 - x^2)}}{25} = frac{sqrt{353x^2 + 40000}}{25}. ] 5. **Solving for x and y:** From 353x^2 + 40000 = 62500, we solve for x: [ 353x^2 = 22500 implies x^2 = frac{22500}{353} approx 63.74. ] [ y^2 = 625 - 63.74 implies y approx 23.60. ] 6. **Calculating [ABC] and [ABCD]:** [ [ABC] = frac{1}{2} cdot x cdot y approx frac{1}{2} cdot 7.98 cdot 23.60 = 94.20. ] [ [ABCD] = [ABC] + [ACD] = 94.20 + 500 = 594.20. ] Conclusion: The new problem provides a plausible scenario for computing the area involving similar triangles, the Pythagorean theorem, and properties of quadrilaterals. The final answer is boxed{594}

answer:We begin by finding the slope (m) of the line that goes through the points (1,3) and (3,11). The formula for the slope between two points ((x_1, y_1)) and ((x_2, y_2)) is: [ m = frac{y_2 - y_1}{x_2 - x_1} = frac{11 - 3}{3 - 1} = frac{8}{2} = 4. ] With the slope m = 4, we use one of the points to find the y-intercept (b). Using the point (1, 3): [ y = mx + b implies 3 = 4(1) + b. ] [ b = 3 - 4 = -1. ] Thus, the sum m+b is: [ m + b = 4 + (-1) = 3. ] So, m+b = boxed{3}.

answer:Given that the sum of the first n terms of the sequence {a_n} is S_n = 4n^2 - n - 8, we can find the nth term a_n by calculating the difference between S_n and S_{n-1}. The formula for a_n is: a_n = S_n - S_{n-1} Substituting n = 4 into the formula for S_n gives: S_4 = 4(4)^2 - 4 - 8 = 64 - 4 - 8 = 52 Substituting n = 3 into the formula for S_n gives: S_3 = 4(3)^2 - 3 - 8 = 36 - 3 - 8 = 25 Therefore, the fourth term a_4 is: a_4 = S_4 - S_3 = 52 - 25 = 27 Thus, a_4 = boxed{27}.